Abstract
We obtain a full resolution result for minimizers in the exterior isoperimetric problem with respect to a compact obstacle in the large volume regime \(v\rightarrow \infty \). This is achieved by the study of a Plateau-type problem with a free boundary (both on the compact obstacle and at infinity), which is used to identify the first obstacle-dependent term (called isoperimetric residue) in the energy expansion, as \(v\rightarrow \infty \), of the exterior isoperimetric problem. A crucial tool in the analysis of isoperimetric residues is a new “mesoscale flatness criterion” for hypersurfaces with bounded mean curvature, which we obtain as a development of ideas originating in the theory of minimal surfaces with isolated singularities.
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1 Introduction
1.1 Overview
Given a compact set \(W\subset \mathbb {R}^{n+1}\) (\(n\ge 1\)), we consider the classical exterior isoperimetric problem associated to W, namely,
in the large volume regime \(v\rightarrow \infty \). Here |E| denotes the volume (Lebesgue measure) of E, and \(P(E;\Omega )\) the (distributional) perimeter of E relative to \(\Omega \), so that \(P(E;\Omega )=\mathcal {H}^n(\Omega \cap \partial E)\) whenever \(\partial E\) is locally Lipschitz. Relative isoperimetric problems are well-known for their analytical [28, Sections 6.4–6.6] and geometric [6, Chapter V] relevance. They are also important in physical applications: beyond the obvious example of capillarity theory [19], exterior isoperimetry at large volumes provides an elegant approach to the Huisken–Yau theorem in general relativity, see [15].
When \(v\rightarrow \infty \), we expect minimizers \(E_v\) in (1.1) to closely resemble balls of volume v. Indeed, by minimality and isoperimetry, denoting by \(B^{(v)}(x)\) the ball of center x and volume v, and with \(B^{(v)}=B^{(v)}(0)\), we find that
Additional information can be obtained by combining (1.2) with quantitative isoperimetry [22, 23]: if \(0<|E|<\infty \), then
The combination of (1.2) and (1.3) shows that minimizers \(E_v\) in \(\psi _W(v)\) are close in \(L^1\)-distance to balls. Based on that, a somehow classical argument exploiting the local regularity theory of perimeter minimizers shows the existence of \(v_0>0\) and of a function \(R_0(v)\rightarrow 0^+\), \(R_0(v)\,v^{1/(n+1)}\rightarrow \infty \) as \(v\rightarrow \infty \), both depending on W, such that, if \(E_v\) is a minimizer of (1.1) with \(v>v_0\), then (see Fig. 1)
here \(\omega _m\) stands for the volume of the unit ball in \(\mathbb {R}^m\), \(B_r(x)\) is the ball of center x and radius r in \(\mathbb {R}^{n+1}\), and \(B_r=B_r(0)\). The picture of the situation offered by (1.2) and (1.4) is thus incomplete under one important aspect: it offers no information related to the specific “obstacle” W under consideration—in other words, two different obstacles are completely unrecognizable from (1.2) and (1.4) alone.
Quantitative isoperimetry gives no information on how W affects \(\psi _W(v)\) for v large
The first step to obtain obstacle-dependent information on \(\psi _W\) is studying \(L^1_\textrm{loc}\)-subsequential limits F of exterior isoperimetric sets \(E_v\) as \(v\rightarrow \infty \). Since the mean curvature of \(\partial E_v\) has order \(v^{-1/(n+1)}\) as \(v\rightarrow \infty \) in \(\Omega \), each \(\partial F\) is easily seen to be a minimal surface in \(\Omega \). A finer analysis leads to establish a more useful characterization of such limits F as minimizers in a “Plateau’s problem with free boundary on the obstacle and at infinity”, whose negative is precisely defined in (1.10) below and denoted by \(\mathcal {R}(W)\). We call \(\mathcal {R}(W)\) the isoperimetric residue of W because it captures the “residual effect” of W in (1.2), as expressed by the limit identity
The study of the geometric information about W stored in \(\mathcal {R}(W)\) is particularly interesting: roughly, \(\mathcal {R}(W)\) is close to an n-dimensional sectional area of W, although its precise value is elusively determined by the behavior of certain “plane-like” minimal surfaces with free boundary on W. The proof of (1.5) itself requires proving a blowdown result for such exterior minimal surfaces, and then extracting sharp decay information towards hyperplane blowdown limits. In particular, in the process of proving (1.5), we shall prove the existence of a positive \(R_2\) (depending on n and W only) such that for every maximizer F of \(\mathcal {R}(W)\), \((\partial F){\setminus } B_{R_2}\) is the graph of a smooth solution to the minimal surfaces equation. An application of Allard’s regularity theorem [3] leads then to complement (1.4) with the following “local” resolution formula: for every \(S>R_2\) and large v in terms of n, W and S,
Interestingly, this already fine analysis gives no information on \(\partial E_v\) in the mesoscale region \(B_{R_0(v)\,v^{1/(n+1)}}\setminus B_S\) between the resolution formulas (1.4) and (1.6). To address this issue, we are compelled to develop what we have called a mesoscale flatness criterion for hypersurfaces with bounded mean curvature. This kind of statement is qualitatively novel with respect to the flatness criteria typically used in the study of blowups and blowdowns of minimal surfaces—although it is clearly related to those tools at the mere technical level—and holds promise for applications to other geometric variational problems. In the study of the exterior isoperimetric problem, it allows us to prove the existence of positive constants \(v_0\) and \(R_1\), depending on n and W only, such that if \(v>v_0\) and \(E_v\) is a minimizer of \(\psi _W(v)\), then
The key difference between (1.6) and (1.7) is that the domain of resolution given in (1.7) overlaps with that of (1.4): indeed, \(R_0(v)\rightarrow 0^+\) as \(v\rightarrow \infty \) implies that \(R_0\,v^{1/(n+1)}< R_1\,v^{1/(n+1)}\) for \(v>v_0\). As a by-product of this overlapping and of the graphicality of \(\partial F\) outside of \(B_{R_2}\), we deduce that boundaries of exterior isoperimetric sets, outside of \(B_{R_2}\), are diffeomorphic to n-dimensional disks. Finally, when \(n\le 6\), and maximizers F of \(\mathcal {R}(W)\) have locally smooth boundaries in \(\Omega \), (1.7) can be propagated up to the obstacle itself; see Remark 1.7 below.
Concerning the rest of this introduction: In Sect. 1.2 we present our analysis of isoperimetric residues, see Theorem 1.1. In Sect. 1.3 we gather all our results concerning exterior isoperimetric sets with large volumes, see Theorem 1.6. Finally, we present the mesoscale flatness criterion in Sect. 1.4 and the organization of the paper in Sect. 1.5.
1.2 Isoperimetric Residues
To define \(\mathcal {R}(W)\) we introduce the class
of those pairs \((F,\nu )\) with \(\nu \in \mathbb {S}^n\) (\(=\) the unit sphere of \(\mathbb {R}^{n+1}\)) and \(F\subset \mathbb {R}^{n+1}\) a set of locally finite perimeter in \(\Omega \) (i.e., \(P(F;\Omega ')<\infty \) for every \(\Omega '\subset \subset \Omega \)), with boundary \(\partial F\) contained in a slab around \(\nu ^\perp =\{x:x\cdot \nu =0\}\) and projecting fully over \(\nu ^\perp \) itself (see Remark 1.5 below): i.e., for some \(\alpha ,\beta \in \mathbb {R}\),
where \(\textbf{p}_{\nu ^\perp }(x)=x-(x\cdot \nu )\,\nu \), \(x\in \mathbb {R}^{n+1}\). In correspondence to W compact, we define the residual perimeter functional, \(\textrm{res}_W:\mathcal {F}\rightarrow \mathbb {R}\cup \{\pm \infty \}\), by
where \({\textbf {C }}_R^\nu =\{x\in \mathbb {R}^{n+1}:|\textbf{p}_{\nu ^\perp }(x)|<R\}\) denotes the (unbounded) cylinder of radius R with axis along \(\nu \)—and where the limsup is actually a monotone decreasing limit thanks to (1.8) and (1.9) (see (4.7) below for a proof). For a reasonably “well-behaved” F, e.g. if \(\partial F\) is the graph of a Lipschitz function over \(\nu ^\perp \), \(\omega _n\,R^n\) is the (obstacle-independent) leading order term of the expansion of \(P(F;{\textbf {C }}_R^\nu \setminus W)\) as \(R\rightarrow \infty \), while \(\textrm{res}_W(F,\nu )\) is expected to capture the first obstacle-dependent “residual perimeter” contribution of \(P(F;{\textbf {C }}_R^\nu \setminus W)\) as \(R\rightarrow \infty \). The isoperimetric residue of W is then defined by maximizing \(\textrm{res}_W\) over \(\mathcal {F}\), so that
see Fig. 2. Clearly \(\mathcal {R}(\lambda \,W)=\lambda ^n\,\mathcal {R}(W)\) if \(\lambda >0\), and \(\mathcal {R}(W)\) is trapped between the areas of the largest hyperplane section and directional projection of W, see (1.11) below. In the simple case when \(n=1\) and W is connected, \(\mathcal {R}(W)=\textrm{diam}\,(W)\) by (1.17) and (1.18) below, although, in general, \(\mathcal {R}(W)\) does not seem to admit a simple characterization, and it is finely tuned to the near-to-the-obstacle behavior of “plane-like” minimal surfaces with free boundary on W. Our first main result collects these (and other) properties of isoperimetric residues and of their maximizers.
If \((F,\nu )\in \mathcal {F}\) then \(\partial F\) is contained in a slab around \(\nu ^\perp \) and is such that \(\partial F\) has full projection over \(\nu ^\perp \). Only the behavior of \(\partial F\) outside W matters in computing \(\textrm{res}_W(F,\nu )\). The perimeter of F in \(\textbf{C}_R^\nu \setminus W\) (depicted as a bold line) is compared to \(\omega _n\,R^n(=\)perimeter of a half-space orthogonal to \(\nu \) in \(\textbf{C}_R^\nu \)); the corresponding “residual” perimeter as \(R\rightarrow \infty \), is \(\textrm{res}_W(F,\nu )\)
Theorem 1.1
(Isoperimetric residues) If \(W\subset \mathbb {R}^{n+1}\) is compact, then there are \(R_2\) and \(C_0\) positive and depending on W with the following property.
(i): If \(\mathcal {S}(W)=\sup \{\mathcal {H}^n(W\cap \Pi ):\Pi \text{ is } \text{ a } \text{ hyperplane } \text{ in } \mathbb {R}^{n+1}\}\) and \(\mathcal {P}(W)=\sup \{\mathcal {H}^n(\textbf{p}_{\nu ^{\perp }}(W)):\nu \in \mathbb {S}^n\}\), then we have
(ii): The family \(\textrm{Max}[\mathcal {R}(W)]\) of maximizers of \(\mathcal {R}(W)\) is non-empty. If \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\), then F is a perimeter minimizer with free boundary in \(\Omega =\mathbb {R}^{n+1}\setminus W\), i.e.
and if \(\mathcal {R}(W)>0\), then \(\partial F\) is contained in the smallest slab \(\{x:\alpha \le x\cdot \nu \le \beta \}\) containing W, and there are \(a,b\in \mathbb {R}\), \(c\in \nu ^\perp \) with \(\max \{|a|,|b|,|c|\}\le C_0\) and \(f\in C^\infty (\nu ^\perp )\) such that
(iii): At fixed diameter, isoperimetric residues are maximized by balls, i.e.
where \(\textrm{cl}\,(X)\) denotes topological closure of \(X\subset \mathbb {R}^{n+1}\). Moreover, if equality holds in (1.15) and \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\), then (1.14) holds with \(b=0\) and \(c=0\), and setting \(\Pi =\big \{y:y\cdot \nu =a\big \}\), we have
for some \(x\in \Pi \). Finally, equality holds in (1.15) if and only if there are a hyperplane \(\Pi \) and a point \(x\in \Pi \) such that
i.e., W contains an \((n-1)\)-dimensional sphere of diameter \(\textrm{diam}\,(W)\), and
Remark 1.2
The assumption \(\mathcal {R}(W)>0\) is quite weak: indeed, if \(\mathcal {R}(W)=0\), then W is purely \(\mathcal {H}^n\)-unrectifiable; see Proposition C.1 in the Appendix. For the role of the topological condition (1.18); see Fig. 3.
The obstacle W (depicted in grey) is obtained by removing a cylinder \(\textrm{C}^{en+1}_{r}\) from a ball \(B_{{d/2}}\) with \(d/2>r\). In this way \(d=\textrm{diam}\,(W)\) and \(B_{d/2}\) is the only ball such that (1.17) can hold. Hyperplanes \(\Pi \) satisfying (1.17) are exactly those passing through the center of \(B_{d/2}\), and intersecting W on a \((n-1)\)-dimensional sphere of radius d/2. For every such \(\Pi \), \(\Omega \setminus (\Pi \setminus B_{d/2})\) has exactly one unbounded connected component, and (1.18) does not hold
Remark 1.3
(Regularity of isoperimetric residues) In the physical dimension \(n=2\), and provided \(\Omega \) has boundary of class \(C^{1,1}\), maximizers of \(\mathcal {R}(W)\) are \(C^{1,1/2}\)-regular up to the obstacle, and smooth away from it. More generally, condition (1.12) implies that \(M=\textrm{cl}\,(\Omega \cap \partial F)\) is a smooth hypersurface with boundary in \(\Omega \setminus \Sigma \), where \(\Sigma \) is a closed set such that \(\Sigma \cap \Omega \) is empty if \(1\le n\le 6\), is locally discrete in \(\Omega \) if \(n=7\), and is locally \(\mathcal {H}^{n-7}\)-rectifiable in \(\Omega \) if \(n\ge 8\); see, e.g. [27, Part III], [30]. Of course, by (1.13), \(\Sigma {\setminus } B_{R_2}=\emptyset \) in every dimension. Moreover, justifying the initial claim concerning the case \(n=2\), if we assume that \(\Omega \) is an open set with \(C^{1,1}\)-boundary, then M is a \(C^{1,1/2}\)-hypersurface with boundary in \(\mathbb {R}^{n+1}\setminus \Sigma \), with boundary contained in \(\partial \Omega \), \(\Sigma \cap \partial \Omega \) is \(\mathcal {H}^{n-3+\varepsilon }\)-negligible for every \(\varepsilon >0\), and Young’s law \(\nu _F\cdot \nu _\Omega =0\) holds on \((M\cap \partial \Omega )\setminus \Sigma \); see, e.g. [13, 14, 24, 25].
Remark 1.4
An interesting open direction is finding additional geometric information on \(\mathcal {R}(W)\), e.g. in the class of convex obstacles. It would also be interesting to quantify more precisely in terms of W some of the other quantities appearing in Theorem 1.1. For instance, it could be that \(R_2 \le C(n) \textrm{diam}\,W\).
Remark 1.5
(Normalization of competitors) We adopt the convention that any set of locally finite perimeter F in \(\Omega \) open is tacitly modified on and by a set of zero Lebesgue measure so to entail \(\Omega \cap \partial F=\Omega \cap \textrm{cl}\,(\partial ^*F)\), where \(\partial ^*F\) is the reduced boundary of F in \(\Omega \); see [27, Proposition 12.19]. Under this normalization, local perimeter minimality conditions like (1.12) (or (3.1) below) imply that \(F\cap \Omega \) is open in \(\mathbb {R}^{n+1}\); see, e.g. [13, Lemma 2.16].
1.3 Resolution of Exterior Isoperimetric Sets
Denoting the family of minimizers of \(\psi _W(v)\) by \(\textrm{Min}[\psi _W(v)]\) and the annulus \(B_s {\setminus } \textrm{cl}\,B_r\) by \(A_r^s\) for \(0<r<s\), our second main result is as follows:
Theorem 1.6
(Resolution of exterior isoperimetric sets) If \(W\subset \mathbb {R}^{n+1}\) is compact, then \(\textrm{Min}[\psi _W(v)]\ne \emptyset \,\,\forall v>0\). Moreover, if \(\mathcal {R}(W)>0\), then
and, depending on n and W only, there are \(v_0\), \(C_0\), \(R_1\), and \(R_2\) positive, and \(R_0(v)\) with \(R_0(v)\rightarrow 0^+\), \(R_0(v)\,v^{1/(n+1)}\rightarrow \infty \) as \(v\rightarrow \infty \), such that, if \(E_v\in \textrm{Min}[\psi _W(v)]\) and \(v>v_0\), then:
(i): There exist \(x\in \mathbb {R}^{n+1}\) and \(u\in C^\infty (\partial B^{(1)})\) such that
where, for any \(G\subset \mathbb {R}^{n+1}\) with locally finite perimeter, \(\nu _G\) is the outer unit normal to G;
(ii): There exist \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\) and \(f\in C^\infty ((\partial F)\setminus B_{R_2})\) with
(iii): \((\partial E_v)\setminus B_{R_2}\) is diffeomorphic to an n-dimensional disk;
(iv): Finally, with (x, u) as in (1.21) and \((F,\nu ,f)\) as in (1.22),
Remark 1.7
(Resolution up to the obstacle) By Remark 1.3 and a covering argument, if \(n\le 6\), \(\delta >0\), and \(v>v_0(n,W,\delta )\), then (1.22) holds with \(B_{R_1\,v^{1/(n+1)}}\setminus I_\delta (W)\) in place of \(B_{R_1\,v^{1/(n+1)}}\setminus B_{R_2}\), where \(I_\delta (W)\) is the open \(\delta \)-neighborhood of W. Similarly, when \(\partial \Omega \in C^{1,1}\) and \(n=2\) (and thus \(\Omega \cap \partial F\) is regular up to the obstacle), we can find \(v_0\) (depending on n and W only) such that (1.22) holds with \(B_{R_1\,v^{1/(n+1)}}\cap \Omega \) in place of \(B_{R_1\,v^{1/(n+1)}}\setminus B_{R_2}\), that is, graphicality over \(\partial F\) holds up to the obstacle itself.
Remark 1.8
If W is convex and J is an half-space, then \(\psi _W(v)\ge \psi _J(v)\) for every \(v>0\), with equality for \(v>0\) if and only if \(\partial W\) contains a flat facet supporting an half-ball of volume v; see [5, 21]. Since \(\psi _J(v)=P(B^{(v)})/2^{1/(n+1)}\) and \(\psi _W(v)-P(B^{(v)})\rightarrow -\mathcal {R}(W)\) as \(v\rightarrow \infty \), the bound \(\psi _W(v)\ge \psi _J(v)\) is far from optimal if v is large. Are there stronger global bounds than \(\psi _W\ge \psi _J\) on convex obstacles? Similarly, it would be interesting to quantify the convergence towards \(\mathcal {R}(W)\) in (1.19), or even that of \(\partial E_v\) towards \(\partial B^{(v)}\) and \(\partial F\) (where (1.20) should not to be sharp).
1.4 The Mesoscale Flatness Criterion
We work with with hypersurfaces M whose mean curvature is bounded by \(\Lambda \ge 0\) in an annulus \(B_{1/\Lambda }\setminus \overline{B}_R\), \(R\in (0,1/\Lambda )\). Even without information on M inside \(B_R\) (where M could have a non-trivial boundary, or topology, etc.) the classical proof of the monotonicity formula can be adapted to show the monotone increasing character on \(r\in (R,1/\Lambda )\) of
(here \(x^{TM}=\textrm{proj}_{T_xM}(x)\)); moreover, if \(\Theta _{M,R,\Lambda }\) is constant over \((a,b)\subset (R,1/\Lambda )\), then \(M\cap (B_b\setminus \overline{B}_a)\) is a cone. Since the constant density value corresponding to \(M=H{\setminus } B_R\), H an hyperplane through the origin, is \(\omega _n\) (as a result of a double cancellation which also involves the “boundary term” in \(\Theta _{H\setminus B_R,R,0}\)), we consider the area deficit
which defines a decreasing quantity on \((R,1/\Lambda )\). Here we use the term “deficit”, rather than the more usual term “excess”, since \(\delta _{M,R,\Lambda }\) does not necessarily have non-negative sign (which is one of the crucial property of “excess quantities” typically used in \(\varepsilon \)-regularity theorems, see, e.g., [27, Lemma 22.11]). Recalling that \(A_r^s=B_s\setminus \textrm{cl}\,(B_r)\) if \(s>r>0\), we are now ready to state the following “smooth version” of our mesoscale flatness criterion (see Theorem 2.1 below for the varifold version):
Theorem 1.9
(Mesoscale flatness criterion (smooth version)) If \(n\ge 2\), \(\Gamma \ge 0\), and \(\sigma >0\), then there are \(M_0\) and \(\varepsilon _0\) positive and depending on n, \(\Gamma \) and \(\sigma \) only, with the following property. Let \(\Lambda \ge 0\), \(R\in (0,1/\Lambda )\), and M be a smooth hypersurface with mean curvature bounded by \(\Lambda \) in \(A^{1/\Lambda }_R\), and with
If there is \(s>0\) such that
and
and if, setting,
we have \(R_*>4\,s\) (and thus \(S_*>4\,s\)), then
for a hyperplane K with \(0\in K\) and unit normal \(\nu _K\), and for \(f\in C^1(K)\).
Remark 1.10
(Structure of the statement) The first condition in (1.26) implicitly requires R to be sufficiently small in terms of \(1/\Lambda \), as it introduces a mesoscale s which is both small with respect to \(1/\Lambda \) and large with respect to R. The condition in (1.27) expresses the flatness of M at the mesoscale s in terms of its area deficit. The final key assumption, \(R_*>4\,s\), expresses the requirement that the area deficit does not decrease too abruptly, and stays above \(-\varepsilon _0\) at least up to the scale \(4\,s\). Under these assumptions, graphicality with respect to a hyperplane K is inferred on an annulus whose lower radius s/32 has the order of the mesoscale s, and whose upper radius \(S_*/16\) can be as large as the decay of the area deficit allows (potentially up to \(\varepsilon _0/16\,\Lambda \) if \(R_*=\infty \)), but in any case not too large with respect to \(1/\Lambda \).
Remark 1.11
(Relationship to other flatness criteria) If M is a hypersurface containing the origin, so that, formally speaking, \(R=0\), and the tangent cone of M there is a plane, Theorem 1 reduces to Allard’s theorem [3]. Similarly, if \(\Lambda =0\) and the exterior minimal hypersurface M has a planar tangent cone at infinity, we recover the exterior blow-down results stated in [35, 36]. In particular, although the motivation for Theorem 1 comes from scenarios where both R and \(\Lambda \) are positive, it can also be viewed as a general framework containing as special cases the blow-up and blow-down flatness criteria for hypersurfaces with planar tangent cones.
Remark 1.12
(Sharpness of the statement) The statement is sharp in the sense that for a surface “with bounded mean curvature and non-trivial topology inside a hole”, flatness can only be established on a mesoscale which is both large with respect to the size of the hole and small with respect to the size of the inverse mean curvature. An example is provided by unduloids \(M_\varepsilon \) with waist size \(\varepsilon \) and mean curvature n in \(\mathbb {R}^{n+1}\); see Fig. 4. A “half-period” of \(M_\varepsilon \) is the graph \(\{x+f_\varepsilon (x)\,e_{n+1}:x\in \mathbb {R}^n,\varepsilon<|x|<R_\varepsilon \}\) of
where \(\varepsilon \) and \(R_\varepsilon \) are the only solutions of \(r^{n-1}=r^n-\varepsilon ^n+\varepsilon ^{n-1}\). Clearly \(f_\varepsilon \) solves \(-\textrm{div}\,(\nabla f_\varepsilon /\sqrt{1+|\nabla f_\varepsilon |^2})=n\) with \(f_\varepsilon =0\), \(|\nabla f_\varepsilon |=+\infty \) on \(\{|x|=\varepsilon \}\), and \(|\nabla f_\varepsilon |=+\infty \) on \(\{|x|=R_\varepsilon \}\), where \(R_\varepsilon =1-\textrm{O}(\varepsilon ^{n-1})\); moreover, \(\min |\nabla f_\varepsilon |\) is achieved at \(r=\textrm{O}(\varepsilon ^{(n-1)/n})\), and if \(r\in (a\,\varepsilon ^{(n-1)/n},b\,\varepsilon ^{(n-1)/n})\) for some \(b>a>0\), then \(|\nabla f_\varepsilon |=\textrm{O}_{a,b}(\varepsilon ^{2(n-1)/n})\). Thus, the horizontal flatness of \(M_\varepsilon \) is no smaller than \(\textrm{O}(\varepsilon ^{2(n-1)/n})\), and has that exact order on a scale which is both very large with respect to the hole (\(\varepsilon ^{(n-1)/n}>\!\!\!>\varepsilon \)) and very small with respect to the inverse mean curvature (\(\varepsilon ^{(n-1)/n}<\!\!\!<1\)).
A half-period of an unduloid with mean curvature n and waist size \(\varepsilon \) in \(\mathbb {R}^{n+1}\). By (1.29), the flatness of \(M_\varepsilon \) is no smaller than \(\textrm{O}(\varepsilon ^{2(n-1)/n})\), and is exactly \(\textrm{O}(\varepsilon ^{2(n-1)/n})\) on an annulus sitting in the mesoscale \(\textrm{O}(\varepsilon ^{(n-1)/n})\). This mesoscale is both very large with respect to waist size \(\varepsilon \), and very small with respect to the size of the inverse mean curvature, which is order one
Remark 1.13
(On the application to \(\psi _W(v)\)) Exterior isoperimetric sets \(E_v\) with large volume v have small constant mean curvature of order \(\Lambda =\Lambda _0(n,W)/v^{1/(n+1)}\). We will work with “holes” of size \(R=R_3(n,W)\), for some \(R_3\) sufficiently large with respect to the radius \(R_2\) appearing in Theorem 1.1–(ii), and determined through the sharp decay rates (1.14). The decay properties of F towards \(\{x:x\cdot \nu =a\}\) when \((F,\nu )\) is a maximizer of \(\mathcal {R}(W)\), the \(C^1\)-proximity of \(\partial E\) to \(\partial B^{(v)}(x)\) for \(|x|\approx (\omega _{n+1}/v)^{1/(n+1)}\), and the \(C^1\)-proximity of \(\partial E\) to \(\partial F\) for some optimal \((F,\nu )\) on bounded annuli of the form \(A^{2\,R_3}_{R_2}\) are used in checking that (1.25) holds with \(\Gamma =\Gamma (n,W)\), that \(E_v\) is flat in the sense of (1.27), and, most importantly, that the area deficit \(\delta _{M,R,\Lambda }\) of \(M=(\partial E_v){\setminus } B_{R_3}\) lies above \(-\varepsilon _0\) up to scale \(r=\textrm{O}(v^{1/(n+1)})\) (which is the key information to deduce \(R_*\approx 1/\Lambda \)), and thus obtain overlapping domains of resolutions in terms of \(\partial B^{(v)}(x)\) and \(\partial F\).
Remark 1.14
While Theorem 1.9 seems clearly applicable to other problems, there are situations where one may need to develop considerably finer “mesoscale flatness criteria”. For example, consider the problem of “resolving” almost CMC boundaries undergoing bubbling [9, 11, 12]. When the oscillation of the mean curvature around a constant \(\Lambda \) is small, such boundaries are close to finite unions of mutually tangent spheres of radius \(n/\Lambda \), and can be covered by \(C^1\)-small normal graphs over such spheres away from their tangency points up to distance \(\varepsilon /\Lambda \), with \(\varepsilon =\varepsilon (n)\), and provided the mean curvature oscillation is small in terms of \(\varepsilon \). For propagating flatness up to a distance directly related to the oscillation of the mean curvature, one would need a version of Theorem 1.9 for “double” spherical graphs; in the setting of blowup/blowdown theorems, this would be similar to passing to the harder case of multiplicity larger than one.
Remark 1.15
(Comparison with blowup/blowdown results) From the technical viewpoint, Theorem 1.9 fits into the framework set up by Allard and Almgren in [1] for the study of blowups and blowdowns of minimal surfaces with tangent integrable cones. At the same time, as exemplified by Remark 1.12, Theorem 1.9 really points in a different direction, since it pertains to situations where neither blowup or blowdown limits make sense. Another interesting point is that, in [1], the area deficit \(\delta _{M,R,\Lambda }\) is considered with a sign, non-positive for blowups, and non-negative for blowdowns, see [1, Theorem 5.9(4), Theorem 9.6(4)]. A key insight here is that for hypersurfaces where the deficit changes sign, graphicality obtained through small negative (or positive) deficit nevertheless persists past the scale where \(\delta _{M,R,\Lambda }\) vanishes, and possibly much farther depending on the surface in question; this is actually crucial for obtaining overlapping domains of resolutions in statements like (1.4) and (1.7).
Remark 1.16
(Extension to general minimal cones) Proving Theorem 1.9 in higher codimension and with arbitrary integrable minimal cones should be possible with essentially the same proof presented here. We do not pursue this extension because, first, only the case of hypersurfaces and hyperplanes is needed in studying \(\psi _W(v)\); and, second, in going for generality, one should work in the framework set up by Simon in [33, 35, 37], which, at variance with the simpler Allard–Almgren’s framework used here, allows one to dispense with the integrability assumption. In this direction, we notice that Theorem 1.9 with \(\Lambda =0\) and \(R_*=+\infty \) is a blowdown result for exterior minimal surfaces (see also Theorem 2.1–(ii), (iii)). A blowdown result for exterior minimal surfaces is outside the scope of [1, Theorem 9.6] which pertains to entire minimal surfaces, but it is claimed, with a sketch of proof, on [35, Page 269] as a modification of [35, Theorem 5.5, \(m<0\)]. It should be mentioned that, to cover the case of exterior minimal surfaces, an additional term of the form \(C\int _{\Sigma }(\dot{u}(t))^{2}\) should be added on the right side of assumption [35, 5.3, \(m<0\)]. This additional term seems not to cause difficulties with the rest of the arguments leading to [35, Theorem 5.5, \(m<0\)]. Thus Simon’s approach, in addition to giving the blowdown analysis of exterior minimal surfaces, should also be viable for generalizing our mesoscale flatness criterion.
1.5 Organization of the Paper
In Sect. 2 we prove Theorem 1.9 (actually, its generalization to varifolds, i.e. Theorem 2.1). In Sect. 3 we prove those parts of Theorem 1.6 which follow simply by quantitative isoperimetry (i.e., they do not require isoperimetric residues nor our mesoscale flatness analysis); see Theorem 3.1. Section 4 is devoted to the study of isoperimetric residues and of their maximizers, and contains the proof Theorem 1.1. We also present there a statement, repeatedly used in our analysis, which summarizes some results from [32]; see Proposition 4.1. Finally, in Sect. 5, we prove the energy expansion (1.19) and those parts of Theorem 1.6 left out in Sect. 3 (i.e., statements (ii, iii, iv)). This final Section is, from a certain viewpoint, the most interesting part of the paper: indeed, it is only the detailed examination of those arguments that clearly illustrates the degree of fine tuning of the preliminary analysis of exterior isoperimetric sets and of maximizers of isoperimetric residues which is needed in order to allow for the application of the mesoscale flatness criterion.
2 A Mesoscale Flatness Criterion for Varifolds
In Sect. 2.1 we introduce the class \(\mathcal {V}_n(\Lambda ,R,S)\) of varifolds used to reformulate Theorem 1.9, see Theorem 2.1. In Sects. 2.2–2.3 we present two reparametrization lemmas (2.3, 2.5) and some “energy estimates” (Theorem 2.6) for spherical graphs; in Sect. 2.4 we state the monotonicity formula in \(\mathcal {V}_n(\Lambda ,R,S)\) and some energy estimates involving the monotonicity gap; in Sect. 2.5, we prove Theorem 2.1.
2.1 Statement of the Criterion
Given an n-dimensional integer rectifiable varifold \(V=\textbf{var}\,(M,\theta )\) in \(\mathbb {R}^{n+1}\), defined by a locally \(\mathcal {H}^n\)-rectifiable set M, and by a multiplicity function \(\theta :M\rightarrow \mathbb {N}\) (see [34]), we denote by \(\Vert V\Vert =\theta \,\mathcal {H}^n\llcorner M\) the weight of V, and by \(\delta V\) the first variation of V, so that \(\delta V(X)=\int \,\textrm{div}\,^T\,X(x)\,dV(x,T)=\int _M\,\textrm{div}\,^M\,X(x)\,\theta \,d\mathcal {H}^n_x\) for every \(X\in C^1_c(\mathbb {R}^{n+1};\mathbb {R}^{n+1})\). Given \(S>R>0\) and \(\Lambda \ge 0\), we consider the family
of those n-dimensional integral varifolds V with \(\textrm{spt}\,V\subset \mathbb {R}^{n+1}\setminus B_R\) and
holds for a Radon measure \(\textrm{bd}_V\) in \(\mathbb {R}^{n+1}\) supported in \(\partial B_R\), and Borel vector fields \(\textbf{H}:\mathbb {R}^{n+1}\rightarrow \mathbb {R}^{n+1}\) with \(|\textbf{H}|\le \Lambda \) and \(\nu _V^\textrm{co}:\partial B_R\rightarrow \mathbb {R}^{n+1}\) with \(|\nu _V^\textrm{co}|=1\). We let \(\mathcal {M}_n(\Lambda ,R,S)=\{V\in \mathcal {V}_n(\Lambda ,R,S):V=\textbf{var}\,(M,1) \text{ for } M \text{ smooth }\}\), that is, \(M\subset \mathbb {R}^{n+1}{\setminus } B_R\) is a smooth hypersurface with boundary in \(A_R^S\), \(\textrm{bdry}\,(M)\subset \partial B_R\), and \(|H_M|\le \Lambda \). If \(V\in \mathcal {M}_n(\Lambda ,R,S)\), then \(\textbf{H}\) is the mean curvature vector of M, \(\textrm{bd}_V=\mathcal {H}^{n-1}\llcorner \textrm{bdry}\,(M)\), and \(\nu _V^\textrm{co}\) is the outer unit conormal to M along \(\partial B_R\). Given \(V\in \mathcal {V}_n(\Lambda ,R,S)\), we define
\(\Theta _{V,R,\Lambda }(r)\) is increasing for \(r\in (R,S)\) (Theorem 2.7–(i) below), and equal to (1.23) when \(V\in \mathcal {M}_n(\Lambda ,R,S)\). The area deficit of V is then defined as in (1.24), while given a hyperplane H in \(\mathbb {R}^{n+1}\) with \(0\in H\) we call the quantity
the angular flatness of V on the annulus \(A_r^s=B_s\setminus \textrm{cl}\,(B_r)\) with respect to H. (See (2.8) for the notation concerning H.)
Theorem 2.1
(Mesoscale flatness criterion) If \(n\ge 2\), \(\Gamma \ge 0\), and \(\sigma >0\) then there are positive constants \(M_0\) and \(\varepsilon _0\), depending on n, \(\Gamma \) and \(\sigma \) only, with the property that: \(\Lambda \ge 0\), \(R\in (0,1/\Lambda )\), \(V\in \mathcal {V}_n(\Lambda ,R,1/\Lambda )\),
and, for some \(s>0\), we have that
then
(i): if \(S_*=\min \{R_*,\varepsilon _0/\Lambda \}<\infty \), then there is an hyperplane \(K\subset \mathbb {R}^{n+1}\) with \(0\in K\) and \(u\in C^1((K\cap \mathbb {S}^n)\times (s/32,S_*/16))\) with
(ii): if \(\Lambda =0\) and \(\delta _{V,R,0}\ge -\varepsilon _0\) on \((s/8,\infty )\), then \(\delta _{V,R,0}\ge 0\) on \((s/8,\infty )\), (2.5) holds with \(S_*=\infty \), and one has decay estimates, continuous in the radius, of the form
for some \(\alpha (n)\in (0,1)\).
Remark 2.2
In Theorem 2.1, graphicality is formulated in terms of the notion of spherical graph (see Sect. 2.2) which is more natural than the usual notion of “cylindrical graph” in setting up the iteration procedure behind Theorem 2.1. Spherical graphicality in terms of a \(C^1\)-small u as in (2.5) translates into cylindrical graphicality in terms of f as in (1.28) with \(f(x)/|x|\approx u(|x|,\hat{x})\) and \(\nabla _{\hat{x}} f(x)-(f(x)/|x|)\approx |x|\,\partial _r\,u(|x|,\hat{x})\) for \(x\ne 0\) and \(\hat{x}=x/|x|\); see, in particular, Lemma B.1 in “Appendix B”.
2.2 Spherical Graphs
We start setting up some notation. We denote by
the family of the oriented hyperplanes \(H\subset \mathbb {R}^{n+1}\) with \(0\in H\), so that for any \(H\in \mathcal {H}\) a unit normal vector \(\nu _H\) to H is defined. Given \(H\in \mathcal {H}\), we set
for the equatorial sphere defined by H on \(\mathbb {S}^n\) and for the orthogonal projections of \(\mathbb {R}^{n+1}\) onto H and onto \(H^\perp =\{t\,\nu _H:t\in \mathbb {R}\}\). We set
Clearly there is \(\sigma _0=\sigma _0(n)>0\) such that if \(H\in \mathcal {H}\) and \(u\in \mathcal {X}_{\sigma _0}(\Sigma _H)\), then
defines a diffeomorphism of \(\Sigma _H\) into an hypersurface \(\Sigma _H(u)\subset \mathbb {S}^n\), namely
We call \(\Sigma _H(u)\) a spherical graph over \(\Sigma _H\). Exploiting the fact that \(\Sigma _H\) is a minimal hypersurface in \(\mathbb {S}^n\) and that if \(\{\tau _i\}_i\) is a local orthonormal frame on \(\Sigma _H\) then \(\nu _H\cdot \nabla _{\tau _i}\tau _j=0\), a second variation computation (see, e.g., [16, Lemma 2.1]) gives, for \(u\in \mathcal {X}_\sigma (\Sigma _H)\),
(where \(n\,\omega _n=\mathcal {H}^{n-1}(\Sigma _H)=\mathcal {H}^{n-1}(\Sigma _H(0))\)). We recall that \(u\in L^2(\Sigma _H)\) is a unit norm Jacobi field of \(\Sigma _H\) (i.e., a zero eigenvector of \(\Delta ^{\Sigma _H}+(n-1)\,\textrm{Id}\,\) with unit \(L^2(\Sigma _H)\)-norm) if and only if there is \(\tau \in \mathbb {S}^n\) with \(\tau \cdot \nu _H=0\) and \(u(\omega )=c_0(n)\,(\omega \cdot \tau )\) (\(\omega \in \Sigma _H\)) for \(c_0(n)=(n/\mathcal {H}^{n-1}(\Sigma _H))^{1/2}\). We denote by \(E^0_{\Sigma _H}\) the orthogonal projection operator of \(L^2(\Omega )\) onto the span of the Jacobi fields of \(\Sigma _H\). The following lemma provides a way to reparameterize spherical graphs over equatorial spheres so that the projection over Jacobi fields is annihilated.
Lemma 2.3
There exist constants \(C_0\), \(\varepsilon _0\) and \(\sigma _0\), depending on the dimension n only, with the following properties:
(i): if \(H,K\in \mathcal {H}\), \(|\nu _H-\nu _K|\le \varepsilon <\varepsilon _0\), and \(u\in \mathcal {X}_\sigma (\Sigma _H)\) for \(\sigma <\sigma _0\), then the map \(T_u^K:\Sigma _H\rightarrow \Sigma _K\) defined by
is a diffeomorphism between \(\Sigma _H\) and \(\Sigma _K\), and \(v_u^K:\Sigma _K\rightarrow \mathbb {R}\) defined by
is such that
(ii): if \(H\in \mathcal {H}\) and \(u\in \mathcal {X}_{\sigma _0}(\Sigma _H)\), then there exist \(K\in \mathcal {H}\) with \(|\nu _H-\nu _K|<\varepsilon _0\) and \(v\in \mathcal {X}_{C_0\,\sigma _0}(\Sigma _K)\) such that
Remark 2.4
It may seem unnecessary to present a detailed proof of Lemma 2.3, as we are about to do, given that, when \(\Sigma _H\) is replaced by a generic integrable minimal surface \(\Sigma \) in \(\mathbb {S}^n\), similar statements are found in the first four sections of [1, Chapter 5]. However, two of those statements, namely [1, 5.3(4), 5.3(5)], seem not to be correct; and the issue requires clarification, since those statements are used in the iteration arguments for the blowup/blowdown theorems [1, Theorem 5.9/Theorem 9.6]; see, e.g., the second displayed chain of inequalities on [1, Page 254]. To explain this issue we momentarily adopt the notation of [1]. In [1, Chapter 5] they consider a family of minimal surfaces \(\{M_t\}_{t\in U}\) in \(\mathbb {S}^n\) obtained as diffeomorphic images of a minimal surface \(M=M_0\). The parameter t ranges in an open ball \(U\subset \mathbb {R}^j\), where j is the dimension of the space of Jacobi fields of M. Given a vector field Z in \(\mathbb {S}^n\), defined on and normal to \(M_t\), they denote by \(F_t(Z)\) the diffeomorphism of \(M_t\) into \(\mathbb {S}^n\) obtained by combining Z with the exponential map of \(\mathbb {S}^n\) (up to lower than second order corrections in Z, this is equivalent to taking the graph of Z over \(M_t\), and then projecting it back on \(\mathbb {S}^n\), which is what we do, following [33], in (2.9)). Then, in [1, 5.2(2)], they define \(\Lambda _t\) as the family of those Z such that \(\textrm{Image}(F_t(Z))=\textrm{Image}(F_0(W))\) for some vector field W normal to M, and, given \(t,u\in U\) and \(Z\in \Lambda _t\), they define \(F_t^u:\Lambda _t\rightarrow \Lambda _u\) as the map between such classes of normal vector fields with the property that \(\textrm{Image}(F_t(Z))=\textrm{Image}(F_u(F_t^u(Z)))\): in particular, \(F_t^u(Z)\) is the vector field that takes \(M_u\) to the same surface to which Z takes \(M_t\). With this premise, in [1, 5.3(5)] they say that if \(t,u\in U\), and \(Z\in \Lambda _t\), then
for a constant C depending on M only. Testing this with \(Z=0\) (notice that \(0\in \Lambda _t\) by [1, 5.3(1)]) one finds \(F_t^u(0)=0\), and thus \(M_t=\textrm{Image}(F_t(0))=\textrm{Image}(F_u(F_t^u(0)))=\textrm{Image}(F_u(0))=M_u\). In particular, \(M_u=M_t\) for every \(t,u\in U\), that is, \(\{M_t\}_{t\in U}\) consists of a single surface, M itself. But this is never the case since \(\{M_t\}_{t\in U}\) always contains, to the least, every sufficiently small rotation of M in \(\mathbb {S}^n\). An analogous problem is contained in [1, 5.3(4)]. Coming back to our notation, the analogous estimate to (2.17) in our setting would mean that, for every \(H,K\in \mathcal {H}\) with \(|\nu _K-\nu _H|<\varepsilon _0\) and \(u\in \mathcal {X}_{\sigma _0}(\Sigma _H)\), \(v_u^K\) defined in (2.10) satisfies
which again gives a contradiction if \(u=0\). A correct estimate, analogous in spirit to (2.18) and still sufficiently precise to be used in iterations, is (2.12) in Lemma 2.3. There should be no obstructionFootnote 1 in adapting our proof to the more general context of integrable cones, and then in using the resulting generalization of (2.12) to implement the iterations needed in [1, Theorem 5.9, Theorem 9.6].
Proof of Lemma 2.3
The constants \(\varepsilon _0\) and \(\sigma _0\) in the statement will be such that \(\sigma _0=\varepsilon _0/C_*\) for a sufficiently large dimension dependent constant \(C_*\).
Step one: To prove statement (i), let \(H,K\in \mathcal {H}\), \(|\nu _H-\nu _K|\le \varepsilon <\varepsilon _0\) and \(u\in \mathcal {X}_{\sigma }(\Sigma _H)\) with \(\sigma <\sigma _0\). Setting (for \(\omega \in \Sigma _H\) and \(x\in \mathbb {R}^{n+1}\setminus \{0\}\))
we have \(T_u^K=\Phi \circ g_u^K\), and, if u is identically 0,
By \(|\textbf{p}_K\nu _H|^2=1-(\nu _H\cdot \nu _K)^2\le 2\,(1-(\nu _H\cdot \nu _K))=|\nu _H-\nu _K|^2\),
In particular, \(|g_u^K|\ge 1-\sigma _0\,\varepsilon _0\ge 1/2\), and since \(\Phi \) and \(\nabla \Phi \) are Lipschitz continuous on \(\{|x|\ge 1/2\}\), we find
Similarly, since \(\omega \cdot \nu _K=\omega \cdot (\nu _K-\nu _H)\) for \(\omega \in \Sigma _H\), we find that
and we thus conclude that \(T_u^K\) is a diffeomorphism between \(\Sigma _H\) and \(\Sigma _K\). As a consequence, the definition (2.10) of \(v_u^K\) is well-posed, and (2.11) immediately follows (in particular, \(\Sigma _H(u)=\Sigma _K(v_u^K)\) is deduced easily from (2.10) and (2.9)). Finally, if we set \(F_u^K(\omega )=v_u^K(T_u^K(\omega ))^2\,J^{\Sigma _H}\,T_u^K(\omega )\) (\(\omega \in \Sigma _H\)), then
where, using again \(|\omega \cdot \nu _K|\le |\nu _H-\nu _K|\) for every \(\omega \in \Sigma _H\), we find
and thus, (2.12), thanks to
Step two: We prove (ii). If \(E_{\Sigma _H}^0[u]=0\), then we conclude with \(K=H\), \(v=u\). We thus assume \(\gamma ^2=\int _{\Sigma _H}\,(E_{\Sigma _H}^0[u])^2>0\), and pick an orthonormal basis \(\{\phi _H^i\}_{i=1}^n\) of \(L^2(\Sigma _H)\cap \{E_{\Sigma _H}^0=0\}\) with \(E_{\Sigma _H}^0[u]=\gamma \,\phi _H^1\) and \(\gamma =\int _{\Sigma _H}u\,\phi _H^1\ne 0\). This corresponds to choosing an orthonormal basis \(\{\tau _H^i\}_{i=1}^n\) of H such that
for \(c_0(n)=(n/\mathcal {H}^{n-1}(\Sigma _H))^{1/2}\). For each \(K\in \mathcal {H}\) with \(\textrm{dist}_{\mathbb {S}^n}(\nu _H,\nu _K)<\varepsilon _0\) we define an orthonormal basis \(\{\tau _K^i\}_{i=1}^n\) of K by parallel transport of \(\{\tau _H^i\}_{i=1}^n\subset H\equiv T_{\nu _H}\mathbb {S}^n\) to \(K\equiv T_{\nu _K}\mathbb {S}^n\). The maps \(\nu \mapsto \tau ^i(\nu ):=\tau _{K(\nu )}^i\) define an orthonormal frame \(\{\tau ^i\}_{i=1}^n\) of \(\mathbb {S}^n\) on the open set \(A=B_{\varepsilon _0}^{\mathbb {S}^n}(\nu _H)=\{\nu \in \mathbb {S}^n:\textrm{dist}_{\mathbb {S}^n}(\nu ,\nu _H)<\varepsilon _0\}\). We denote by \(\rho _H^K\) the rotation of \(\mathbb {R}^{n+1}\) which takes H into K by setting \(\rho _H^K(\tau _H^i)=\tau _K^i\) and \(\rho _H^K(\nu _H)=\nu _K\). By the properties of parallel transport we have that
Finally, we define an \(L^2(\Sigma _K)\)-orthonormal basis \(\{\phi _K^i\}_{i=1}^n\) of \(L^2(\Sigma _K)\cap \{E_{\Sigma _K}^0=0\}\) by setting \(\phi _K^i(\omega )=c_0(n)\,\omega \cdot \tau _K^i\) (\(\omega \in \Sigma _K\)), and correspondingly consider the map \(\Psi _u:A\rightarrow \mathbb {R}^n\) defined by setting
where \(v_u^{K(\nu )}\) is well-defined for every \(\nu \in A\) thanks to step one. We now claim the existence of \(\nu _*\in A\) such that
Before proving (2.22), we use it to deduce (2.13)–(2.16), thus finishing the proof of (ii) and the lemma modulo (2.22). With \(K=K(\nu _*)\) and \(v=v_u^K\) we deduce (2.13) from (2.11) and (2.14) from \(\Psi _u(\nu _*)=0\). By (2.26) and (2.27), if \(\eta =\textrm{dist}_{\mathbb {S}^n}(\nu _*,\nu _H)\), then
that is (2.15). Finally, (2.16) follows from (2.15) and (2.12).
Turning now towards proving (2.22), by the area formula, (2.10), and \(\textbf{q}_{K(\nu )}[e]=\nu \cdot e\), we find that
so that (2.19) gives that
By definition of A and by (2.20) and (2.21),
where \(\Psi _*:A\rightarrow \mathbb {R}^n\) is defined by \(e_j\cdot \Psi _*(\nu )=c_0(n)\,\int _{\Sigma _H} (\nu \cdot \omega )\,(\tau _H^j\cdot \omega )\,d\mathcal {H}^{n-1}_\omega \) (\(\nu \in A\)). Recalling that \(\{\tau ^i\}_{i=1}^n\) is an orthonormal frame of \(\mathbb {S}^n\) on A, with \(\nabla _{\tau ^i}\nu =\tau ^i(\nu )=\tau _{K(\nu )}^i=\rho _H^{K(\nu )}[\tau ^i_H]\), we find that
By (2.21), (2.23) and (2.24) we conclude that
Let us finally consider the map \(h:A\times [0,1]\rightarrow \mathbb {R}^n\),
which defines an homotopy between \(\Psi _*\) and \(\Psi _u\). By (2.25) and (2.26) we see that if \(\nu \in \partial A\), that is, if \(\textrm{dist}_{\mathbb {S}^n}(\nu ,\nu _H)=\varepsilon _0\), then, denoting by \([\nu _H,\nu ]_s\) the unit-speed length minimizing geodesic from \(\nu _H\) to \(\nu \), considering that \([\nu _H,\nu ]_s\in A\) for every \(s\in (0,\varepsilon _0\)), and that \(\mathbb {S}^n\) is close to be flat in A, we find
provided \(\sigma _0=\varepsilon _0/C_*\) is small enough with respect to \(\varepsilon _0\) (i.e., provided \(C_*\) is large), \(\varepsilon _0\) is small in terms of \(c_0\), and where we have used \(\Psi _*(\nu _H)=0\) and
to deduce \(|h_t(\nu _H)|\le C(n)\,\sigma _0\). This proves that \(0\not \in \partial \,h_t(\partial A)\) for every \(t\in [0,1]\), so that \(\deg (h_t,A,0)\) is independent of \(t\in [0,1]\). In particular, \(h_0=\Psi _u\) and \(h_1=\Psi _*\) give \(\deg (\Psi _u,A,0)=\deg (\Psi _*,A,0)=1\), where we have used \(\Psi _*(\nu _H)=0\) and the fact that, up to decreasing the value of \(\varepsilon _0\), \(\Psi _*\) is injective on A. By \(\deg (\Psi _u,A,0)=1\), there is \(\nu _*\in A\) such that \(\Psi _u(\nu _*)=0\), as claimed in (2.22). \(\square \)
2.3 Energy Estimates for Spherical Graphs Over Annuli
Given \(H\in \mathcal {H}\) and \(0<r_1<r_2\) we let \(\mathcal {X}_\sigma (\Sigma _H,r_1,r_2)\) be the class of those \(u\in C^1(\Sigma _H\times (r_1,r_2))\) such that, setting \(u_r=u(\cdot ,r)\), one has \(u_r\in \mathcal {X}_\sigma (\Sigma _H)\) for every \(r\in (r_1,r_2)\) and \(|r\,\partial _r u|\le \sigma \) on \(\Sigma _H\times (r_1,r_2)\). If \(u\in \mathcal {X}_\sigma (\Sigma _H,r_1,r_2)\), then the spherical graph of u over \(A_{r_1}^{r_2} \cap H\), given by
is an hypersurface in \(A_{r_1}^{r_2}\). It is useful to keep in mind that \(\Sigma _H(0,r_1,r_2)=\{r\,\omega :\omega \in \Sigma \,,r\in (r_1,r_2)\}=H\cap A_{r_1}^{r_2}\) is a flat annular region of area \(\omega _n\,(r_2^n-r_1^n)\), and that if \(\sigma <\sigma _1=\sigma _1(n)\), then
Lemma 2.5
There are \(\varepsilon _0\), \(\sigma _0\), \(C_0\) positive, depending on n only, such that:
(i): if \(H,K\in \mathcal {H}\), \(\nu _H\cdot \nu _K>0\), \(|\nu _H-\nu _K|=\varepsilon <\varepsilon _0\), \(u\in \mathcal {X}_{\sigma }(\Sigma _H,r_1,r_2)\), and \(\sigma <\sigma _0\), then there is \(v\in \mathcal {X}_{C_0(\sigma +\varepsilon )}(\Sigma _H,r_1,r_2)\) such that \(\Sigma _K(v,r_1,r_2)=\Sigma _H(u,r_1,r_2)\).
(ii): if \(H\in \mathcal {H}\), \(u\in \mathcal {X}_{\sigma _0}(\Sigma _H,r_1,r_2)\), and \((a,b)\subset \subset (r_1,r_2)\), then there exist \(K\in \mathcal {H}\), \(v\in \mathcal {X}_{C_0\,\sigma _0}(\Sigma _K,r_1,r_2)\), and \(r_*\in [a,b]\) such that
Moreover, for every \(r\in (r_1,r_2)\),
Proof
We prove statement (i). If \(|\nu _H-\nu _K|=\varepsilon <\varepsilon _0\), since \(u_r\in \mathcal {X}_{\sigma }(\Sigma _H)\) for every \(r\in (r_1,r_2)\), by Lemma 2.3–(i) we see that \(T_r:\Sigma _H\rightarrow \Sigma _K\),
is a diffeomorphism between \(\Sigma _H\) and \(\Sigma _K\), and \(v_r:\Sigma _K\rightarrow \mathbb {R}\),
satisfies \(v_r\in \mathcal {X}_{C_0\,(\sigma +\varepsilon )}(\Sigma _K)\), \(\Sigma _H(u_r)=\Sigma _K(v_r)\) for every \(r\in (r_1,r_2)\), and
Since \(u\in \mathcal {X}_{\sigma }(\Sigma _H,r_1,r_2)\), and \(T_r\) and \(v_r\) depend smoothly on \(u_r\), setting \(v(\omega ,r):=v_r(\omega )\) we have \(\Sigma _H(u,r_1,r_2)=\Sigma _K(v,r_1,r_2)\) (by \(\Sigma _H(u_r)=\Sigma _K(v_r)\) for every \(r\in (r_1,r_2)\)), and \(v\in \mathcal {X}_{C_0\,(\sigma +\varepsilon )}(\Sigma _H,r_1,r_2)\) (\(|r\,\partial _rv_r|\le C_0(\sigma +\varepsilon )\) is deduced by differentiation in (2.31) and (2.32), and by \(|u_r|,|r\,\partial _ru_r|<\sigma \)).
Step two: We prove (ii). Let \(\gamma =\min _{\rho \in [a,b]}\int _{\Sigma _H}\,\big (E_{\Sigma _H}^0[u_\rho ]\big )^2\), and let \(r_*\in [a,b]\) be such that the minimum \(\gamma \) is achieved at \(r=r_*\). If \(\gamma =0\), then we set \(K=H\) and \(v=u\). If \(\gamma >0\), then we apply Lemma 2.3–(ii) to \(u_{r_*}\in \mathcal {X}_{\sigma _0}(\Sigma _H)\), and find \(K\in \mathcal {H}\) with \(|\nu _K-\nu _H|<\varepsilon _0\) and \(v_{r_*}\in \mathcal {X}_{C_0\,s_0}(\Sigma _K)\) such that \(\Sigma _H(u_{r_*})=\Sigma _K(v_{r_*})\) and
Since \(v_{r_*}=v(\cdot ,r_*)\) for v constructed in step one starting from u, H and K, we deduce (2.30) by (2.33) and (2.35), while (2.34) is (2.29). \(\square \)
We will use two basic “energy estimates” for spherical graphs over annuli. To streamline the application of these estimates to diadic families of annuli we consider intervals \((r_1,r_2)\) and \((r_3,r_4)\) are \((\eta ,\eta _0)\)-related, meaning that
for some \(\eta _0>\eta >0\), and with \(r_0=(r_1+r_2)/2=(r_3+r_4)/2\); in particular, \((r_3,r_4)\) is contained in, and concentric to, \((r_1,r_2)\). The case \(\Lambda =0\) of the following statement is the codimension one, equatorial spheres case of [1, Lemma 7.14, Theorem 7.15].
Theorem 2.6
(Energy estimates for spherical graphs) If \(n\ge 2\) and \(\eta _0>\eta >0\), then there are \(\sigma _0=\sigma _0(n,\eta _0,\eta )\) and \(C_0=C_0(n,\eta _0,\eta )\) positive, with the following property. If \(H\in \mathcal {H}\), \(\Lambda \ge 0\), and \(u\in \mathcal {X}_{\sigma }(\Sigma _H,r_1,r_2)\) is such that \(\max \{1,\Lambda \,r_2\}\,\sigma \le \sigma _0\) and \(\Sigma _H(u,r_1,r_2)\) has mean curvature bounded by \(\Lambda \) in \(A_{r_1}^{r_2}\), then, whenever \((r_1,r_2)\) and \((r_3,r_4)\) are \((\eta ,\eta _0)\)-related as in (2.36),
Moreover, if there is \(r\in (r_1,r_2)\) s.t. \(E_{\Sigma _H}^0 u_r=0\) on \(\Sigma _H\), then we also have
Proof
Since this proof is quite long and the arguments are not needed to understand the rest of the paper, we postpone it to “Appendix A”. \(\square \)
2.4 Monotonicity for Exterior Varifolds with Bounded Mean Curvature
The following theorem states the monotonicity of \(\Theta _{V,R,\Lambda }\) for \(V\in \mathcal {V}_n(\Lambda ,R,S)\), and provides, when V corresponds to a spherical graph, a quantitative lower bound for the gap in the associated monotonicity formula; the case \(\Lambda =0\), \(R=0\) is contained in [1, Lemma 7.16, Theorem 7.17].
Theorem 2.7
(i): If \(V\in \mathcal {V}_n(\Lambda ,R,S)\), then
(ii): There is \(\sigma _0(n)\) such that, if \(V\in \mathcal {V}_n(\Lambda ,R,S)\) and, for some \(H\in \mathcal {H}\), \(u\in \mathcal {X}_\sigma (\Sigma ,r_1,r_2)\) with \(\sigma \le \sigma _0(n)\), and \((r_1,r_2)\subset (R,S)\), we have
then
(iii): Finally, given \(\eta _0>\eta >0\), there exist \(\sigma _0\) and \(C_0\) depending on n, \(\eta _0\), and \(\eta \) only, such that if the assumptions of part (i) and part (ii) hold and, in addition to that, we also have \(\max \{1,\Lambda \,r_2\}\,\sigma \le \sigma _0\) and
then, whenever \((r_1,r_2)\) and \((r_3,r_4)\) are \((\eta ,\eta _0)\)-related as in (2.36), we have
Proof
We give details of the proof of (i) when \(V\in \mathcal {M}_n(\Lambda ,R,S)\) (whereas the general case is addressed as in [34, Section 17]). By the coarea formula, the divergence theorem and \(|\textbf{H}|\le \Lambda \), for a.e. \(\rho >R\),
where \(\textrm{Mon}(V,\rho )=(d/d\rho )\int _{B_\rho {\setminus } B_R}\,|x^\perp |^2\,|x|^{-n-2}\,d\Vert V\Vert \). Since \(\textrm{Mon}(V,\rho )\ge 0\), this proves (i). Assuming now (2.37), a straightforward computation which we omit (c.f. for example in [1, Lemma 3.5(6), Lemma 7.16]), we see that, under (2.37),
thus proving (ii). To prove (iii), we set \(a=r_0\,(1-(\eta +\eta _0)/2)\) and \(b=r_0\,(1+(\eta +\eta _0)/2)\), so that (a, b) and \((r_3,r_4)\) are \((\eta ,(\eta +\eta _0)/2)\)-related, and \((r_1,r_2)\) and (a, b) are \(((\eta +\eta _0)/2,\eta _0)\)-related (in particular, \((r_3,r_4)\subset (a,b)\subset (r_1,r_2)\)). By suitably choosing \(\sigma _0\) in terms of n, \(\eta \) and \(\eta _0\), we can apply Theorem 2.6 with \((r_3,r_4)\) and (a, b), so to find (with \(C=C(n,\eta _0,\eta )\))
Thanks to (2.39) we can apply Theorem 2.6 with (a, b) and \((r_1,r_2)\) to find
We find (2.40) by (2.38) and \((\Lambda \,b)^2\,(b^n-a^n)\le (\Lambda \,r_2)^2\,r_2^n\). \(\square \)
2.5 Proof of the Mesoscale Flatness Criterion
As a final preliminary result to the proof of Theorem 2.1, we prove the following lemma, where Allard’s regularity theorem is combined with a compactness argument to provide the basic graphicality criterion used throughout the iteration. The statement should be compared to [1, Lemma 5.7].
Lemma 2.8
(Graphicality lemma) Let \(n\ge 2\). For every \(\sigma >0\), \(\Gamma \ge 0\), \((\lambda _3,\lambda _4)\subset \subset (\lambda _1,\lambda _2)\subset \subset (0,1)\), and \((\eta _1,\eta _2)\subset \subset (0,1)\), there are positive constants \(\varepsilon _1\) and \(M_1\), depending only on n, \(\sigma \), \(\Gamma \), \((\lambda _1,\lambda _2)\), \((\lambda _3,\lambda _4)\), and \((\eta _1,\eta _2)\), and \(\varepsilon _2\) and \(M_2\), depending only on n, \(\sigma \), \(\Gamma \), \(\lambda _1\), and \((\eta _1,\eta _2)\), with the following properties.
(i): If \(\Lambda \ge 0\), \(R\in (0,1/\Lambda )\), \(V\in \mathcal {V}_n(\Lambda ,R,1/\Lambda )\),
there exists \(r>0\) such that
and if, for some \(K\in \mathcal {H}\), we have
then there exists \(u\in \mathcal {X}_\sigma (\Sigma _K,\eta _1 \, r,\eta _2 \, r)\) such that
(ii): If \(\Lambda \), R, and V are as in (i), (2.42) holds, and there exists r such that
then there exists \(K\in \mathcal {H}\) and \(u\in \mathcal {X}_\sigma (\Sigma _K, \eta _1\, r, \eta _2\, r)\) such that
Proof
Step one: As a preliminary, we first show that if V is a stationary, n-dimensional, integer rectifiable varifold in \(B_1\) such that
for some \(K\in \mathcal {H}\) and \(0<\beta _1<\beta _2\le 1\), then \(V = \textbf{var}\,(K\cap B_1,\left. 1\right| _{K\cap B_1})\).
Let \(\beta '\in (\beta _1,\beta _2)\) and \(\varphi _1\), \(\varphi _2\in C^\infty (\mathbb {R}^{n+1};[0,1])\) be such that \(\textrm{spt}\, \varphi _1 \subset B_{\beta _2}\), \(\left. \varphi _1\right| _{B_{\beta '}}\equiv 1\), and \(\varphi _1+\varphi _2\equiv 1\). As a consequence of (2.49) and the stationarity of V in \(B_{\beta _2}\), for \(X\in C_c^1(\mathbb {R}^{n+1}{\setminus } (K \cap (\overline{B}_{\beta _2} {\setminus } B_{\beta '}))\), we have
Then by the convex hull property [34, Theorem 19.2], 
. By the constancy theorem [34, Theorem 41.1], 
for some constant \(\theta \). Furthermore, since V assigns non-trivial mass to \(B_{\beta _2}\) by (2.49) and is integer rectifiable, \(\theta \ge 1\). Therefore \(0\in \textrm{spt}\Vert V\Vert \), and the monotonicity formula gives \(\omega _n \le \lim _{r\rightarrow 0^+}\Vert V\Vert (B_r)r^{-n} \le \Vert V\Vert (B_1)\le \omega _n\). Thus V is a stationary, n-dimensional, integer rectifiable varifold in \(B_1\) with constant area ratios \(\omega _n\) and \(\textrm{spt}V \cap A_{\beta _1}^{\beta _2}\subset K\), so \(V = \textbf{var}\,(K\cap B_1,\left. 1\right| _{K\cap B_1})\).
Step two: We prove item (i) by contradiction. If it were false, we could find \(\sigma >0\), \(\Gamma \ge 0\), \((\lambda _3,\lambda _4)\subset \subset (\lambda _1,\lambda _2)\subset \subset (0,1)\), \((\eta _1,\eta _2)\subset (0,1)\), with \(K_j\in H\), positive numbers \(R_j\), \(\Lambda _j<1/R_j\), \(r_j\), and \(W_j\in \mathcal {V}_n(\Lambda _j,R_j,1/\Lambda _j)\) such that \(\Vert W_j\Vert \big (A_{\lambda _3\,r_j}^{\lambda _4\,r_j}\big )>0\), \(\Vert \textrm{bd}_{W_j}\Vert (\partial B_{R_j})\le \Gamma \,R_j^{n-1}\), \(\Vert W_j\Vert (B_\rho \setminus B_{R_j})\le \Gamma \,\rho ^n\) for every \(\rho \in (R_j,1/\Lambda _j)\), and \(\rho _j=R_j/r_j\rightarrow 0\), \(r_j\,\Lambda _j\rightarrow 0\), \(\delta _{W_j,R_j,\Lambda _j}(r_j)\rightarrow 0\), and \(r_j^{-n}\int _{B_{\lambda _2\,r_j}{\setminus } B_{\lambda _1\,r_j}} \omega _{K_j}^2\,d\Vert W_j\Vert \rightarrow 0\), but there is no \(u\in \mathcal {X}_{\sigma }(\Sigma _{K_j},\eta _1\,r_j,\eta _2\,r_j)\) with the property that \(W_j\) corresponds to \(\Sigma _{K_j}(u,\eta _1\,r_j,\eta _2\,r_j)\) on \(A_{\eta _1\,r_j}^{\eta _2\,r_j}\). Hence, setting \(V_j=W_j/r_j\), no \(u\in \mathcal {X}_{\sigma }(\Sigma _{K_j},\eta _1,\eta _2)\) can exist such that \(V_j\) corresponds to \(\Sigma _{K_j}(u,\eta _1,\eta _2)\) on \(A_{\eta _1}^{\eta _2}\), despite the fact that each \(V_j\) belongs to \(\mathcal {V}_n(r_j\,\Lambda _j,\rho _j,1/(r_j\,\Lambda _j))\) and satisfies
Clearly we can find \(K\in \mathcal {H}\) such that, up to extracting subsequences, \(K_j\cap B_1\rightarrow K\cap B_1\) in \(L^1(\mathbb {R}^{n+1})\). Similarly, by (2.50), we can find an n-dimensional integer rectifiable varifold V such that \(V_j\rightharpoonup V\) as varifolds in \(B_1\setminus \{0\}\). Since the bound on the distributional mean curvature of \(V_j\) on \(B_{1/(\Lambda _j\,r_j)}{\setminus } \overline{B}_{\rho _j}\) is \(r_j\,\Lambda _j\), and since \(\rho _j\rightarrow 0^+\) and \(r_j\,\Lambda _j\rightarrow 0^+\), it also follows that V is stationary in \(B_1\setminus \{0\}\), and thus, by a standard argument and since \(n\ge 2\), on \(B_1\). By \(\Vert V_j\Vert (A_{\lambda _3}^{\lambda _4})>0\), for every j there is \(x_j\in A_{\lambda _3}^{\lambda _4}\cap \textrm{spt}\,V_j\), so that, up to extracting subsequences, \(x_j\rightarrow x_0\) for some \(x_0\in \overline{A}_{\lambda _3}^{\lambda _4}\cap \textrm{spt}\,V\). By \((\lambda _3,\lambda _4)\subset \subset (\lambda _1,\lambda _2)\), there is \(\rho >0\) such that \(B_\rho (x_0)\subset A_{\lambda _1}^{\lambda _2}\), hence
thus proving \(V\,\llcorner \, A_{\lambda _1}^{\lambda _2}\ne \emptyset \). By this last fact, by \(\omega _K=0\) on \((\textrm{spt}\,V)\cap A_{\lambda _1}^{\lambda _2}\), and by the constancy theorem [34, Theorem 41.1], we have
At the same time, since \(\Vert \textrm{bd}_{V_j}\Vert (\partial B_{\rho _j})\le \Gamma \,\rho _j^{n-1}\) and \(\Vert V_j\Vert (B_\rho {\setminus } B_{\rho _j})\le \Gamma \,\rho ^n\) for every \(\rho \in (\rho _j,1/(\Lambda _j\,r_j))\supset (\rho _j\,1)\), by (2.50),
Since V is stationary in \(B_1\) and integer rectifiable, and since (2.51) and (2.52) imply (2.49) with \(\lambda _1=\beta _1\) and \(\lambda _2=\beta _2\), the first step yields \(V = \textbf{var}\,(K\cap B_1,\left. 1\right| _{K\cap B_1})\). By Allard’s regularity theorem and by \(V_j\rightharpoonup V\) as \(j\rightarrow \infty \) we deduce the existence of a sequence \(\{u_j\}_j\), with \(u_j\in \mathcal {X}_{\sigma _j}(\Sigma _K,\eta _1,\eta _2)\) for some \(\sigma _j\rightarrow 0\) as \(j\rightarrow \infty \), such that \(V_j\) corresponds to \(\Sigma _K(u_j,\eta _1,\eta _2)\) in \(A_{\eta _1}^{\eta _2}\) for j large enough. As soon as j is large enough to give \(\sigma _j<\sigma \), we have reached a contradiction.
Step three: For item (ii), we again argue by contradiction. Should the lemma be false, then we could find \(\sigma >0\), \(\Gamma \ge 0\), \(\lambda _1\in (0,1)\), \((\eta _1,\eta _2)\subset (0,1)\), positive numbers \(R_j\), \(\Lambda _j<1/R_j\), \(r_j\), and, by the same rescaling as in step two, \(V_j\in \mathcal {V}_n(r_j\,\Lambda _j,\rho _j,1/(r_j\,\Lambda _j))\) with
such that there exists no \(u\in \mathcal {X}_{\sigma }(\Sigma _{K_j},\eta _1,\eta _2)\) with the property that \(V_j\) corresponds to \(\Sigma _{K_j}(u,\eta _1,\eta _2 )\) on \(A_{\eta _1}^{\eta _2 }\). As in step two, we can find an n-dimensional integer rectifiable varifold \(V{=\textbf{var}\,(M,\theta )}\) such that \(V_j\rightharpoonup V\) as varifolds in \(B_1{\setminus }\{0\}\) and V is stationary on \(B_1\). If for some \(K\in \mathcal {H}\), \(V =\textbf{var}\,(K\cap B_1,\left. 1\right| _{K\cap B_1})\), then using Allard’s theorem as in the proof of (i), we have a contradiction. So we prove \(V =\textbf{var}\,(K\cap B_1,\left. 1\right| _{K\cap B_1})\).
For every \(r\in [\lambda _1,1]\), using \(\rho _j\rightarrow 0^+\) and \(r_j \Lambda _j \rightarrow 0^+\) in conjunction with (2.53), and then the monotonicity of \(\delta _{V_j, \rho _j, r_j\,\Lambda _j}\) and (2.54), we have
Thus the convergence \(V_j \rightharpoonup V\) and the monotonicity of \(\Vert V \Vert (B_r)/r^n\) yield
By (2.55), 
for some locally \(\mathcal {H}^n\)-rectifiable cone \(C\subset \mathbb {R}^{n+1}\) and zero homogeneous \(\theta _C:C\rightarrow \mathbb {N}\). Now since the integer rectifiable varifold cone \(\textbf{var}\,(C,\theta _C)\) is stationary in \(B_1\setminus \overline{B}_{\lambda _1}\), it is stationary in \(\mathbb {R}^{n+1}\) by \(n\ge 2\), and due to (2.55), it satisfies \(\int _{C\cap B_1} \theta _C\, d\mathcal {H}^n = \omega _n\). Therefore \(C =K\) for some \(K\in \mathcal {H}\), and \(\theta _C\equiv 1\). From the definition of C, it follows that
Finally, (2.55) and (2.56) give (2.49) with \(\beta _1=\lambda _1\), \(\beta _2=1\). The result of step one then completes the proof that \(V =\textbf{var}\,(K\cap B_1,\left. 1\right| _{K\cap B_1})\). \(\square \)
Proof of Theorem 2.1
The proof proceeds in four steps, which we outline here. Precise statements can be found at the beginning of each step. First, we assume that \(\delta _{V,R,\Lambda }(s/8)\ge 0\), and prove that \(C^1\)-graphicality can be propagated from s/32 to an upper radius \(S_+/16\le S_*/16\) as long as \(\delta _{V,R,\Lambda }(S_+)\) remains non-negative and \(S_+\le \varepsilon _0/\Lambda \). This is then enough to prove the exterior blow-down result in part (ii) of Theorem 2.1 in step two. In the third step, we argue that if \(\delta _{V,R,\Lambda }(s/8)\le 0\), then \(C^1\)-graphicality can be propagated inwards from \(S_*/2\) down to s/32. The details in this step are quite similar to the first, so we summarize them. Finally, the first and third steps are combined in step four to conclude the proof Theorem 2.1–(i), in which there are no sign restrictions on the deficit.
Step one: In this step, given \(n\ge 2\), \(\Gamma \ge 0\), and \(\sigma >0\), we prove the existence of \(\varepsilon _0\) and \(M_0\) (specified below in (2.65) and (2.66), and depending on n, \(\Gamma \), and \(\sigma \)) such that if (2.1), (2.2), (2.3) and (2.4) hold with \(\varepsilon _0\) and \(M_0\), and in addition
then there exist \(K_+\in \mathcal {H}\) and \(u_+\in \mathcal {X}_\sigma (\Sigma _{K_+},s/32,S_{+}/16 )\) such that
where
We start by imposing some constraints on the constants \(\varepsilon _0\) and \(M_0\). For the finite set
we let \(\sigma _0=\sigma _0(n)\) be such that Lemma 2.5–(ii), Theorems 2.6, and 2.7–(ii), (iii) hold for every \((\eta _0,\eta )\in J\), Lemma 2.5–(i) holds for \(\sigma <\sigma _0\), and
we shall henceforth assume, without loss of generality, that
Moreover, for \(\varepsilon _1\) and \(M_1\) as in Lemma 2.8–(i) and \(C_0\) as in Lemma 2.5, we let
We also assume that
where \(C(n,\Gamma )\) will be specified in (2.96)–(2.97), \(C_0\) is as in Lemma 2.5, and \(\varepsilon _0\) is smaller than both of the n-dependent \(\varepsilon _0\)’s appearing in Lemmas 2.3 and 2.5. Lastly, we choose \(\overline{\sigma }>0\) such that
and then, for \(\varepsilon _2\), \(M_2\) as in Lemma 2.8–(ii), we choose \(\varepsilon _0\) and \(M_0\) so that
Let us now recall that, by assumption, \(V\in \mathcal {V}_n(\Lambda ,R,1/\Lambda )\) is such that
in particular, by Theorem 2.7–(i),
Moreover, we are assuming the existence of s with \(\max \{64,M_0\}\,R<s<\varepsilon _0/4\,\Lambda \) such that
so that the latter inequality, together with (2.59), implies
By (2.68), (2.69) and (2.70) we have
By (2.67), the specification of s satisfying (2.2), and (2.71), the assumptions (2.42), (2.47), and (2.48), respectively, of Lemma 2.8–(ii) with \(r=s\), \(\lambda _1=1/8\), and \((\eta _1,\eta _2)=(1/32,1/2)\) are satisfied due to our choices (2.65) and (2.66). Setting \(H_0=H\), where \(H\in \mathcal {H}\) is from the application of Lemma 2.8–(ii), we thus find \(u_0\in \mathcal {X}_{\overline{\sigma }}(\Sigma _{H_0},s/32,s/2)\) such that
If it is the case that \(S_+=4\,s\), we are in fact done with the proof of (2.58), since then \(s/2\ge S_+ / 16\). We may for the rest of this step assume then that \(S_+ > 4\,s\), so that
First, we observe that thanks to (2.72) and then (2.64),
We let \(s_j=2^{j-3}\,s\) for \(j\in \mathbb {Z}_{\ge -1}\). By (2.73) and by \(s<\varepsilon _0/4\,\Lambda \le \varepsilon _0'/4\,\Lambda \) there exists \(N\in \{j\in \mathbb {N}:j\ge 2\}\cup \{+\infty \}\) such that
Notice that if \(\Lambda >0\) then it must be \(N<\infty \). We are now in the position to make the following:
Claim: There exist \(\tau =\tau (n)\in (0,1)\) and \(\{(H_j,u_j)\}_{j=0}^{N-2}\) with \(H_j\in \mathcal {H}\) such that, setting
for every \(j=0,...,N-2\),
where \(C_0\) is from Lemma 2.5, and
additionally, for every \(j=1,...,N-2\),
Proof of the claim: We argue by induction. Clearly (2.76)\(_{j=0}\), (2.77)\(_{j=0}\), (2.78)\(_{j=0}\) and (2.79)\(_{j=0}\) are, respectively, (2.72), (2.69) and (2.74). This concludes the proof of the claim if \(N=2\), therefore we shall assume \(N\ge 3\) for the rest of the argument. To set up the inductive argument, we consider \(\ell \in \mathbb {N}\) such that: either \(\ell =0\); or \(1\le \ell \le N-3\) and (2.76), (2.77), (2.78), and (2.79) hold for \(j=0,...,\ell \), and (2.80), (2.81) and (2.82) hold for \(j=1,...,\ell \); and prove that all the conclusions of the claim hold with \(j=\ell +1\).
The validity of (2.78)\(_{j=\ell +1}\) is of course immediate from (2.71) and (2.75). Also, after proving (2.82)\(_{j=\ell +1}\), we will be able to combine it with (2.78)\(_{j=\ell +1}\) and (2.75) to deduce (2.79)\(_{j=\ell +1}\). We now prove, in order, (2.80), (2.76), (2.77), (2.81), and (2.82) with \(j=\ell +1\).
To prove (2.80)\(_{j=\ell +1}\): Let \([a,b]\subset \subset (s_\ell ,s_{\ell +1})\) with \((b-a)=(s_{\ell +1}-s_\ell )/2\), so that
Keeping in mind (2.76)\(_{j=\ell }\), (2.77)\(_{j=\ell }\), we can apply Lemma 2.5–(ii) with \((r_1,r_2)=(s/32,4\,s_\ell )\) and \([a,b]\subset (s_\ell ,s_{\ell +1})\) to find \(H_{\ell +1}\in \mathcal {H}\),
(with \(C_0\) as in Lemma 2.5–(ii)) and
such that, thanks also to (2.83),
In particular, (2.87) is (2.80)\(_{j=\ell +1}\).
To prove (2.76)\(_{j=\ell +1}\) and (2.77)\(_{j=\ell +1}\): Notice that (2.84), (2.85) do not imply (2.76)\(_{j=\ell +1}\) and (2.77)\(_{j=\ell +1}\), since, in (2.77)\(_{j=\ell +1}\), we are claiming the graphicality of V inside \(A_{s/32}^{4\,s_{\ell +1}}\) (which is strictly larger than \(A_{s/32}^{4\,s_\ell }\)), and in (2.76)\(_{j=\ell +1}\) we are claiming that \(u_{\ell +1}\) has \(C^1\)-norm bounded by \(\sigma \) or \(\sigma /2\,C_0\) (depending on the radius), and not just by \(C_0\,\sigma _0\) (with \(C_0\) as in Lemma 2.5–(ii)).
We want to apply Lemma 2.8–(i) with \(K=H_{\ell +1}\) and
We check the validity of (2.43), (2.44), (2.45), and (2.46) with \(\varepsilon _1=\varepsilon _0'\) and \(M_1=M_0'\) for these choices of r, \(\lambda _1\), \(\lambda _2\), \(\lambda _3\), \(\lambda _4\), \(\eta _1\), \(\eta _2\), and K. Since \(r=8\,s_{\ell +1}\ge s\ge \max \{M_0,64\,R\}\ge \max \{M_0',64\,R\}\), and since (2.75) and \(\ell +1\le N\) give \(r=8\,s_{\ell +1}\le \varepsilon _0/\Lambda \le \varepsilon _0'/\Lambda \), we deduce the validity of (2.43) with \(r=8\,s_{\ell +1}\). The validity of (2.44) with \(r=8\,s_{\ell +1}\) is immediate from (2.71) by our choice (2.62) of \(\varepsilon _0'\). Next we notice that
thanks to (2.77)\(_{j=\ell }\), so that (2.45) holds for r, \(\lambda _3\) and \(\lambda _4\) as in (2.89). Finally, by (2.28) (which can be applied to \(u_{\ell +1}\) thanks to (2.61)), (2.85) and (2.76)\(_{j=\ell }\), and, then by (2.88), we have
where in the last inequality we have used (2.79)\(_{j=\ell }\). Again by our choice (2.62) of \(\varepsilon _0'\), we deduce that (2.46) holds with r, \(\lambda _1\) and \(\lambda _2\) as in (2.89). We can thus apply Lemma 2.8–(i), and find \(v\in \mathcal {X}_{\sigma /2\,C_0}(\Sigma _{H_{\ell +1}},s_{\ell +1}/4,4\,s_{\ell +1})\) such that
By (2.85), (2.77)\(_{j=\ell }\), and (2.90), \(v=u_{\ell +1}\) on \(\Sigma _{H_{\ell +1}}\times (s_{\ell +1}/4, 4\,s_\ell )\). We can thus use v to extend \(u_{\ell +1}\) from \(\Sigma _{H_{\ell +1}}\times (s/32, 4\,s_\ell )\) to \(\Sigma _{H_{\ell +1}}\times (s/32,4\,s_{\ell +1})\), and, thanks to (2.85), (2.77)\(_{j=\ell }\) and (2.90), the resulting extension is such that
The bound (2.91) is part of (2.76)\(_{j=\ell +1}\), and (2.92) is (2.77)\(_{j=\ell +1}\), so in order to complete the proof of (2.76)\(_{j=\ell +1}\) and (2.77)\(_{j=\ell +1}\), it remains to show that the \(C^1\)-norm of u is bounded by \(\sigma \) in between s/32 and \(4\,s_{\ell }\).
Towards this end, we record the following consequence of taking square roots in (2.81)\(_{j=m}\) (using \(\delta _{V,R,\Lambda }\ge 0\) from (2.75)) and summing over \(m=1,...,i\) for any \(1\le i \le \ell \): for \(\alpha =\sum _{k=0}^\infty 2^{-k/2}\) and \(\tilde{C}(n,\Gamma )=\tau ^{1/2}(1+\Gamma )\),
where in the last line we have used (2.75) and (2.71). By induction, utilizing (2.57), (2.65) for the base case and (2.93) for the induction step we have
Now by the positivity of \(\delta _{V,R,\Lambda }\) and (2.82)\(_{j=\ell }\), for all \(m=1,...,\ell \),
In turn, by (2.80)\(_{j=\ell +1}\), (2.74) and (2.95), then (2.75) and (2.94)\(_{i=\ell -1}\),
for a suitable \(C(n,\Gamma )\). We use (2.96) to see
Now \(u_i \in \mathcal {X}_{\sigma /2\,C_0}(\Sigma _{H_j},s_i/4, 4\,s_i)\) by (2.76)\(_{j=i}\), and \(\sigma /2\,C_0\) and \(|\nu _{H_i}-\nu _{H_{\ell +1}}|\) are small enough to apply Lemma 2.5–(i) by our choice of \(\sigma \) above (2.61) and (2.97) with (2.63), respectively. Then we obtain \(w_i\) corresponding to V on \(A_{s_i/4}^{4\,s_i}\) and in \( \mathcal {X}_{\sigma /2 + C_0|\nu _{H_i}-\nu _{H_{\ell +1}}|}(\Sigma _{H_{\ell +1}}, s_i/4,4\,s_i)\), and by (2.97), (2.63),
so \(w_i\in \mathcal {X}_\sigma (\Sigma _{H_{\ell +1}},s_i/4,4\,s_i)\). Finally, since they represent the same surface over \(\Sigma _{H_{\ell +1}}\), \(w_i=u_{\ell +1}\) on \(A^{4\,s_i}_{s_i/4}\). Gathering these estimates for \(i=0,...,\ell \), we have \(u_{\ell +1}\in \mathcal {X}_\sigma (\Sigma _{H_{\ell +1}},s/32, 4\,s_\ell )\), which finishes the proof of (2.76)\(_{j=\ell +1}\).
To prove (2.81)\(_{j=\ell +1}\): We set \(r_0=(s_\ell +s_{\ell +1})/2\) and notice that for \(\eta _0=1/3\),
For \(\eta =1/6\) we correspondingly set
and notice that \((\eta _0,\eta )\in J\), see (2.60). With the aim of applying Theorem 2.7–(iii) to these radii, we notice that (2.77)\(_{j=\ell +1}\) implies that assumption (2.37) holds with \(H=H_{\ell +1}\) and \(u=u_{\ell +1}\), while, by (2.86), \(r=s_\ell ^*\in (s_\ell ,s_{\ell +1})\) is such that (2.39) holds. By \(\Lambda \,s_{\ell +1}\le \varepsilon _0\le 1\), (2.75), and (2.40), with \(C(n)=C_0(n,1/6,1/3)\) for \(C_0\) as in Theorem 2.7–(iii), we have
Setting for brevity \(\delta =\delta _{V,R,\Lambda }\) and \(\Theta =\Theta _{V,R,\Lambda }\), and recalling that
we have
By \(\Lambda \,s_\ell \le 1\) and since \(s_{\ell }^+\le s_\ell \le \varepsilon _0/8\,\Lambda \) thanks to \(\ell <N\), we can use the upper bound \(\Vert V\Vert (B_\rho \setminus B_R)\le \Gamma \,\rho ^n\) with \(\rho \in (R,s_\ell ^+)\subset (R,1/\Lambda )\), to find that
for a constant \(C_*(n)\). By rearranging terms and using the monotonicity of \(\delta \) on \((R,\infty )\) and \((s_\ell ^-,s_\ell ^+)\subset (s_\ell ,s_{\ell +1})\) we find that
We finally notice that by (2.98), (2.99), \(\eta _0=1/3\), and \(\eta =1/6\), we have
so that we find that \(\delta (s_{\ell +1})\le \tau \{\delta (s_\ell )+(1+\Gamma )\,\Lambda \,s_\ell \}\) (i.e. (2.81)\(_{j=\ell +1}\)) with
To prove (2.82)\(_{j=\ell +1}\): We finally prove (2.82)\(_{j=\ell +1}\), i.e.
By (2.77)\(_{j=\ell +1}\) we know that
Now, (2.36) holds with \(r_0=3\,s_\ell \) and \((\eta _0,\eta )=(2/3,1/3)\in J\), see (2.60), if
Since \(s_\ell ^*\in (s_\ell ,s_{\ell +1})\subset (r_1,r_2)\), by (2.101), (2.86) and \((r_1,r_2)\subset (s/32,4\,s_{\ell +1})\) we can apply Theorem 2.6 to deduce that
Again by (2.101), Theorem 2.7–(ii) with \((r_1,r_2)=(s_\ell ,8\,s_\ell )\) gives
The last two estimates combined give (2.100), which finishes the claim.
Proof of (2.58): We assume \(S_+<\infty \) (that is either \(\Lambda >0\) or \(R_+<\infty \)), and recall that we have already proved (2.58) if \(S_+=4\,s\). Otherwise, N (as defined in (2.75)) is finite, with \(2^N\le \frac{S_+}{s} <2^{N+1}\). By (2.76)\(_{j=N-2}\) and (2.77)\(_{j=N-2}\), we have that \(u_{N-2}\in \mathcal {X}_{\sigma }(\Sigma _{H_{N-2}},s/32,4\,s_{N-2})\) and V corresponds to \(\Sigma _{H_{N-2}}(u_{N-2},s/32,4\,s_{N-2})\) on \(A_{s/32}^{4\,s_{N-2}}\). Since \(4\,s_{N-2} =2^{N+1}\,s/16> S_+/16\), we deduce (2.58) with \(K_+=H_{N-2}\) and \(u_+=u_{N-2}\).
Step two: In this step we prove statement (ii) in Theorem 2.1. We assume that \(\Lambda =0\) and that
where we have set for brevity \(\delta =\delta _{V,R,0}\). We must first show that
Since \(\delta \) is decreasing in r, it has a limit \(\lim _{r\rightarrow \infty }\delta (r)=:\delta _\infty \ge -\varepsilon _0\), and we want to show that \(\delta _\infty =0\). Next, we know that for any sequence \(R_i\rightarrow \infty \), \(V/R_i\) converges locally in the varifold sense to a limiting integer rectifiable varifold cone W. By the local varifold convergence and \(n\ge 2\), W is stationary in \(\mathbb {R}^{n+1}\), and it is the case that
Up to decreasing \(\varepsilon _0\) if necessary (and recalling that \(\delta _{W,0,0}\) is the usual area excess multiplied by \(-1\)), Allard’s theorem and the fact that \(W/r = W\) imply that W corresponds to a multiplicity one plane. In particular, it must be that \(\delta _\infty =0\), which together with the monotonicity of \(\delta \) yields (2.103).
By (2.103), \(S_+=S_*=\infty \), and so by (2.76) and (2.77), there is a sequence \(\{(H_j,u_j)\}_{j=0}^N\) but with \(N=\infty \) now, satisfying
Notice that, in asserting the validity of (2.107) with \(j\ge 1\), we have used (2.103) to estimate \(-\delta (s_{j+2})\le 0\) in (2.82)\(_{j}\). By iterating (2.106) we find
which, combined with (2.107) and (2.105), gives, for every \(j\ge 1\),
thanks also to \(\tau =\tau (n)\) and, again, to (2.103). By (2.110), for every \(j\ge 0\), \(k\ge 1\), we have \(|\nu _{H_{j+k}}-\nu _{H_j}| \le C(n)\,\sqrt{\delta (s/8)}\,\sum _{h=1}^{k+1}\big (\sqrt{\tau }\big )^{j-1+h}\), so that there exists \(K\in \mathcal {H}\) such that
In particular, for j large enough, we have \(\varepsilon _j<\varepsilon _0\), and thus, by Lemma 2.5–(i) and by (2.104) we can find \(v_j\in \mathcal {X}_{C(n)\,(\sigma +\varepsilon _j)}(\Sigma _{K},s/32,4\,s_{j})\) such that
By (2.112), \(v_{j+1}=v_{j}\) on \(\Sigma _{K}\times (s/32,4\,s_{j})\). Since \(s_{j}\rightarrow \infty \) we have thus found \(u\in \mathcal {X}_{C(n)\,\sigma }(\Sigma _{K};s/32,\infty )\) such that
which corresponds to (2.5) with \(\infty \) in place of \(S_*\).
To prove (2.6), we notice that if \(r\in (s_j,s_{j+1})\) for some \(j\ge 1\), then, setting \(\tau =(1/2)^\alpha \) (i.e., \(\alpha =\log _{1/2}(\tau )\in (0,1)\)) and noticing that \(r/s\le 2^{j+1-3}\), by (2.68) and (2.108) we have
where in the last inequality (2.102) was used again; this proves (2.6). To prove (2.7), we recall that \(\omega _K(y)=\arctan (|\nu _K\cdot \hat{y}|/|\textbf{p}_K\,\hat{y}|)\), provided \(\arctan \) is defined on \(\mathbb {R}\cup \{\pm \infty \}\), and where \(\hat{y}=y/|y|\), \(y\ne 0\). Now, by (2.113),
so that \(|\textbf{p}_K\,\hat{y}|\ge 1/2\) for \(y\in (\textrm{spt}V){\setminus } B_{s/32}\); therefore, by (2.111), up to further decreasing the value of \(\varepsilon _0\), and recalling \(\delta (s/8)\le \varepsilon _0\), we conclude
for every \(j\in \mathbb {N}\cup \{+\infty \}\) (if we set \(H_\infty =K\)). By (2.114) we easily find
from which we deduce that, if \(j\ge 1\) and \(r\in (s_j,s_{j+1})\), then
where (2.67) was used to bound \(\Vert V\Vert (A_\rho ^{2\,\rho })\le \Gamma \,(2\,\rho )^n\) with \(\rho =s_j,s_{j+1}\in (R,1/\Lambda )\). By (2.104) we can exploit (2.28) on the first two integrals, so that taking (2.111) into account we find that, if \(j\ge 1\) and \(r\in (s_j,s_{j+1})\), then \(r^{-n}\,\int _{A_r^{2\,r}}\omega _K^2\,d\Vert V\Vert \le C(n)\{T_j+T_{j+1}\big \}+C(n)\,\Gamma \,\tau ^j\,\delta (s/8) \le C(n)\,(1+\Gamma )\,\tau ^j\,\delta (s/8)\), where in the last inequality we have used (2.109). Since \(\tau ^j\le C(n)\,(s/r)^\alpha \), we conclude the proof of (2.7), and thus, of Theorem 2.1–(ii).
Step three: In this step, given \(n\ge 2\), \(\Gamma \ge 0\), and \(\sigma >0\), we claim the existence of \(\varepsilon _0\) and \(M_0\), depending only on n, \(\Gamma \), and \(\sigma \), such that if (2.1), (2.2), (2.3) and (2.4) hold with \(\varepsilon _0\) and \(M_0\), and in addition,
then there exist \(K_-\in \mathcal {H}\) and \(u_-\in \mathcal {X}_\sigma (\Sigma _{K_-},s/32,S_{*}/2 )\) such that
where \(S_*\) and \(R_*\) are as in Theorem 2.1. The argument is quite similar to that of the first step, with minor differences due to the opposite sign of the deficit. The first is that the iteration instead begins at the outer radius \(S_*\) and proceeds inwards via intermediate radii \(s_j=2^{-j}S_*\), and the second is that, in the analogue of the graphicality propagation claims (2.76)\(_{j=\ell +1}\) and (2.77)\(_{j=\ell +1}\), the negative sign on \(\delta _{V,R,\Lambda }\) is used to sum the “tilting” between successive planes \(H_j\) and \(H_{j+1}\).
Step four: Finally, we combine steps one and three to prove statement (i) in Theorem 2.1. Before choosing the parameters \(\varepsilon _0\) and \(M_0\), we need a preliminary result. We claim that for any \(\varepsilon '>0\), there exists \(\sigma '(\varepsilon ')>0\) such that if \(r_1<r_2\), \(K_1\), \(K_2\in \mathcal {H}\) with \(\nu _{K_1}\cdot \nu _{K_2}\ge 0\) and accompanying \(u_i\in \mathcal {X}_{\sigma '}(\Sigma _{K_i}, r_1, r_2)\), and M is a smooth hypersurface such that \(M\cap A_{r_1}^{r_2}\) corresponds to \(\Sigma _{K_i}(u_i,r_1,r_2)\) for \(i=1,2\), then
It is immediate from \(\nu _{K_1}\cdot \nu _{K_2}\ge 0\) and the fact that the \(L^\infty \)-bounds on \(u_i\) imply that M is contained in the intersection of two cones containing \(K_1\) and \(K_2\), whose openings become arbitrarily narrow as \(\sigma '\rightarrow 0\) .
Fix \(n\ge 2\), \(\Gamma \ge 0\), and \(\sigma >0\); we assume without loss of generality that \(\sigma <\sigma _0\), where \(\sigma _0\) is the dimension-dependent constant from Lemma 2.5. We choose \(\varepsilon '\) with corresponding \(\sigma '\) according to (2.117) such that, up to decreasing \(\sigma '\) if necessary,
where \(\varepsilon _0\), \(C_0\) are as in Lemma 2.5. Next, we choose \(\varepsilon _0=\varepsilon _0(n,\Gamma ,\sigma )\) and \(M_0=M_0(n,\Gamma ,\sigma )\) to satisfy several restrictions: first, \(\varepsilon _0\) is smaller than the \(\varepsilon _0\) from Lemma 2.5 and each \(\varepsilon _0(n,\Gamma ,\sigma ')\) from steps one and three, and \(M_0\) is larger than \(M_0(n,\Gamma ,\sigma ')\) from those steps; second, with \(\varepsilon _2\) and \(M_2\) as in Lemma 2.8–(ii), we also assume that
In the remainder of this step, we suppose that
In proving Theorem 2.1–(i), there are three cases depending on whether \(\delta _{V,R,\Lambda }\) changes sign on \([s/8, S_*]\).
Case one: \(\delta _{V,R,\Lambda }(r)\ge 0\) for all \( r\in [s/8, S_*]\). If the deficit is non-negative, then in particular
and \(S_*=S_+\), where \(S_+\) was defined in (2.59). By our choice of \(\varepsilon _0\) and \(M_0\) at the beginning of this step and the equivalence of (2.121) and (2.57), step one applies and the conclusion (2.58) is (2.5). Thus Theorem 2.1–(i) is proved.
Case two: \(\delta _{V,R,\Lambda }(r)\le 0\) for all \(r\in [s/8,S_*]\). Should the deficit be non-positive in this interval, then in particular, (2.115) holds in addition to (2.1), (2.2), (2.3) and (2.4). Therefore, by our choice of \(\varepsilon _0\) and \(M_0\), step three applies. The conclusion (2.116) is (2.5) (in fact with larger upper radii \(S_*/2\)), and Theorem 2.1–(i) is proved.
Case three: \(\delta _{V,R,\Lambda }\) changes sign in \([s/8,S_*]\). By the monotonicity of \(\delta _{V,R,\Lambda }\),
First, by (2.122), (2.57) is satisfied, so (2.58) gives \(K_+\in \mathcal {H}\) and \(u_+\in \mathcal {X}_{\sigma '}(\Sigma _{K_+},s/32,S_{+}/16 )\) such that
where
If \(S_+ = S_*\), then (2.123) is (2.5) and we are done. So we assume for the rest of this case that \(S_+<S_*\), which implies \(S_+\ne \varepsilon _0/\Lambda \) and thus
Next, we make the following
Claim: There exists \(K_-\in \mathcal {H}\) and \(u_-\in \mathcal {X}_{\sigma '}(\Sigma _{K_-},R_+/2,S_{*}/2 )\) such that
Proof of the claim: There are two subcases.
Subcase one: \(16\,R_+ < \varepsilon _0/4\,\Lambda \) and \( 64\,R_+ < R_* \). We claim the conditions of step three are verified at \(s'=16\,R_+\). First, (2.1) holds from (2.120), and
(which is (2.2)) holds due to the assumption of the subcase and \(16R_+\ge s > \max \{64,M_0 \}R\). Next, \(2\, R_+< R_*/4\) by the assumption of the subcase, which combined with the monotonicity of \(\delta _{V,R,\Lambda }\) and (2.124) gives \(-\varepsilon _0\le \delta _{V,R,\Lambda }(2\,R_+) \le 0\). This implies (2.115) and (2.3) with \(s'=16\,R_+\). Lastly, (2.4) holds at \(s'=16\,R_+\) since \(64\,R_+ < R_*\). Thus we apply (2.116) at \(s'=16\,R_+\), finding (2.126).
Subcase two: One or both of \(16\,R_+ \ge \varepsilon _0/4\,\Lambda \), \(64\,R_+ \ge R_*\) hold. In this case,
We wish to apply Lemma 2.8–(ii) with \(r=S_*\), \(\lambda _1=\frac{1}{16}\), \((\eta _1,\eta _2) = (\frac{1}{128},\frac{1}{2})\). By (2.120), (2.42) holds for V, and by (2.119), (2.120), and \(S_*\ge 4\,s\),
which is (2.47). Finally, we have \(R_*\ge S_*/16\ge s/8\), so that by the definition of \(R_*\), (2.3), the monotonicity of \(\delta _{V,R,\Lambda }\), and (2.119),
which is (2.48). By the choices (2.119), Lemma 2.8–(ii) applies and yields the existence of \(K_-\in \mathcal {H}\) and \(u_-\in \mathcal {X}_{\sigma '}(\Sigma _{K_-},S_*/128,S_{*}/2 )\) such that
By (2.127), \(S_*/128 \le R_+/2\), so (2.128) implies (2.126). The proof of the claim is complete.
Returning to the proof of Theorem 2.1–(i) under the assumption (2.122), we recall (2.125) and choose \(R'\in (R_+, \min \{2\,R_+,S_* \})\). Again, we want to apply Lemma 2.8–(ii), this time with \( r=R'\), \(\lambda _1=\frac{1}{16}\), and \((\eta _1,\eta _2) = (1/128,1/2)\). To begin with, V satisfies (2.42) as usual from (2.120). Second, (2.47) holds at \(R'\) by \(s\le R'\le S_*\), (2.120), and the choices (2.119). By the monotonicity of \(\delta _{V,R,\Lambda }\) and \([R'/16,R']\subset [R_+/16, S_*] \subset [s/4, S_*]\), (2.48) is valid by our choice (2.119) of \(\varepsilon _0\). The graphicality result from Lemma 2.8–(ii) therefore yields \(K\in \mathcal {H}\) and \(u\in \mathcal {X}_{\sigma '}(\Sigma _{K},R'/128, R'/2 )\) such that
Now \(s/32\le R'/128< R_+/64<S_+/16\) by \(R' < 2\,R_+\) and (2.125), and \(R_+/2<R'/2 <S_*/2\), so by (2.123) and (2.126), respectively, we have
where \(u_+\in \mathcal {X}_{\sigma '}(\Sigma _{K_+},R'/128, S_+/16)\) and \(u_-\in \mathcal {X}_{\sigma '}(\Sigma _{K_-},R_+/2, R'/2)\). Furthermore, up multiplying \(\nu _{K_+}\) or \(\nu _{K_-}\) by minus one, we may assume \(\nu _{K}\cdot \nu _{K_\pm }\ge 0\). Thus V is represented by multiple spherical graphs on nontrivial annuli. By combining (2.129), (2.1302.131) and (2.1302.131), \(\nu _{K}\cdot \nu _{K_\pm }\ge 0\) and \(\sigma '=\sigma '(\varepsilon ')\), (2.117) applies and gives
But \(\varepsilon '\) was chosen according to (2.118) so that Lemma 2.5–(i) is applicable; that is, since \(\varepsilon '<\varepsilon _0\) and \(\sigma '<\sigma _0\) from that lemma, we may reparametrize (2.123) and (2.126), respectively, as
where
By (2.118), \(C_0(\sigma '+\varepsilon ')\le \sigma \), and by \(R'/128<S_+/16<R_+/2<R'/2\), (2.1322.133) and (2.1322.133), we may extend the u defined in (2.129) onto \(\Sigma _K \times (s/32,S_*/2) \) using \(w_+\) and \(w_-\) with \(C^1\)-norm bounded by \(\sigma \). The resulting extension is such that (2.5) holds, so the proof of Theorem 2.1 is finished. \(\square \)
3 Application of Quantitative Isoperimetry
Here we apply quantitative isoperimetry to prove Theorem 1.6–(i) and parts of Theorem 1.6–(iv).
Theorem 3.1
If \(W\subset \mathbb {R}^{n+1}\) is compact, \(v>0\), then \(\textrm{Min}[\psi _W(v)]\ne \emptyset \). Moreover, depending on n and W only, there are \(v_0\), \(C_0\), \(\Lambda _0\) positive, \(s_0\in (0,1)\), and \(R_0(v)\) with \(R_0(v)\rightarrow 0^+\) and \(R_0(v)\,v^{1/(n+1)}\rightarrow \infty \) as \(v\rightarrow \infty \), such that, if \(v> v_0\) and \(E_v\) is a minimizer of \(\psi _W(v)\), then:
(i): \(E_v\) is a \((\Lambda _0/v^{1/(n+1)},s_0\,v^{1/(n+1)})\)-perimeter minimizer with free boundary in \(\Omega \), that is
for every \(F\subset \Omega =\mathbb {R}^{n+1}{\setminus } W\) with \(E_v\Delta F\subset \subset B_r(z)\) and \(r<s_0\,v^{1/(n+1)}\);
(ii): There exists \(x\in \mathbb {R}^{n+1}\) such that
if \(\mathcal {R}(W)>0\), then there also exists \(u\in C^\infty (\partial B^{(1)})\) such that
(iii): if \(\mathcal {R}(W)>0\) and x and u depend on \(E_v\) as in (3.2) and (3.3), then
Remark 3.2
(Improved convergence) We will repeatedly use the following fact (see, e.g. [7, 8, 18, 20]): If \(\Omega \) is an open set, \(\Lambda \ge 0\), \(s>0\), if \(\{F_j\}_j\) are \((\Lambda ,s)\)-perimeter minimizers in \(\Omega \), i.e. if it holds that
whenever \(G_j\Delta F_j\subset \subset B_r(x)\subset \subset \Omega \) and \(r<s\), and if F is an open set with smooth boundary in \(\Omega \) such that \(F_j\rightarrow F\) in \(L^1_{\textrm{loc}}(\Omega )\) as \(j\rightarrow \infty \), then for every \(\Omega '\subset \subset \Omega \) there is \(j(\Omega ')\) such that
for a sequence \(\{u_j\}_j\subset C^1(\Omega \cap \partial F)\) with \(\Vert u_j\Vert _{C^1(\Omega \cap \partial F)}\rightarrow 0\).
Compare the terminology used in (3.1) and (3.5): when we add “with free boundary”, the “localizing balls” \(B_r(x)\) are not required to be compactly contained in \(\Omega \), and the perimeters are computed in \(B_r(x)\cap \Omega \).
Proof of Theorem 3.1
Step one: We prove \(\textrm{Min}[\psi _W(v)]\ne \emptyset \) for all \(v>0\). Since W is compact, \(B^{(v)}(x)\subset \subset \Omega \) for |x| large. Hence there is \(\{E_j\}_j\) with
for every \(F\subset \Omega \) with \(|F|=v\). Hence, up to extracting subsequences, \(E_j\rightarrow E\) in \(L^1_\textrm{loc}(\mathbb {R}^{n+1})\) with \(P(E;\Omega ) \le \varliminf _{j\rightarrow \infty }P(E_j;\Omega )\), where \(E\subset \Omega \) and \(|E|\le v\). We now make three remarks concerning E:
(a): If \(\{\Omega _i\}_{i\in I}\) are the connected components of \(\Omega \), then \(\Omega \cap \partial ^*E=\emptyset \) if and only if \(E = \bigcup _{i\in I_0} \Omega _i\) (\(I_0\subset I\)). Indeed, \(\Omega \cap \partial ^*E=\emptyset \) implies \(\textrm{cl}\,(\partial ^*E)\cap \Omega =\partial E\cap \Omega \), hence \(\partial E\subset \partial \Omega \) and \(E = \bigcup _{i\in I_0} \Omega _i\). The converse is immediate.
(b): If \(\Omega \cap \partial ^*E\ne \emptyset \), then we can construct a system of “volume–fixing variations” for \(\{E_j\}_j\). Indeed, if \(\Omega \cap \partial ^*E\ne \emptyset \), then there are \(B_{S_0}(x_0)\subset \subset \Omega \) with \(P(E;\partial B_{S_0}(x_0))=0\) and a vector field \(X\in C^\infty _c(B_{S_0}(x_0);\mathbb {R}^{n+1})\) such that \(\int _E\textrm{div}\,\,X=1\). By [27, Theorem 29.14], there are constants \(C_0,c_0>0\), depending on E itself, with the following property: whenever \(|(F\Delta E)\cap B_{S_0}(x_0)|<c_0\), then there is a smooth function \(\Phi ^F:\mathbb {R}^n\times (-c_0,c_0)\rightarrow \mathbb {R}^n\) such that, for each \(|t|<c_0\), the map \(\Phi _t^F=\Phi ^F(\cdot ,t)\) is a smooth diffeomorphism with \(\{\Phi _t^F\ne \textrm{id}\,\}\subset \subset B_{S_0}(x_0)\), \(|\Phi _t^F(F)|=|F|+t\), and \(P(\Phi _t^F(F);B_{S_0}(x_0))\le (1+C_0\,|t|)\,P(F;B_{S_0}(x_0))\). For j large enough, we evidently have \(|(E_j\Delta E)\cap B_{S_0}(x_0)|<c_0\), and thus we can construct smooth functions \(\Phi ^j:\mathbb {R}^n\times (-c_0,c_0)\rightarrow \mathbb {R}^n\) such that, for each \(|t|<c_0\), the map \(\Phi _t^j=\Phi ^j(\cdot ,t)\) is a smooth diffeomorphism with \(\{\Phi _t^j\ne \textrm{id}\,\}\subset \subset B_{S_0}(x_0)\), \(|\Phi _t^j(E_j)|=|E_j|+t\), and \(P(\Phi _t^j(E_j);B_{S_0}(x_0))\le (1+C_0\,|t|)\, P(E_j;B_{S_0}(x_0))\).
(c): If \(\Omega \cap \partial ^*E\ne \emptyset \), then E is bounded. Since \(|E|\le v<\infty \), it is enough to prove that \(\Omega \cap \partial ^*E\) is bounded. In turn, taking \(x_0\in \Omega \cap \partial ^*E\), and since W is bounded and \(|E|<\infty \), the boundedness of \(\Omega \cap \partial ^*E\) descends immediately by the following density estimate: there is \(r_1>0\) such that
To prove (3.7), let \(r_1>0\) be such that \(|B_{r_1}|<c_0\), let x and r be as in (3.7), and set \(F_j=(\Phi _t^j(E_j)\cap B_{S_0}(x_0))\cup [E_j{\setminus }(B_r(x)\cup B_{S_0}(x_0))]\) for \(t=|E_j\cap B_r(x)|\) (which is an admissible value of t by \(|B_{r_1}|<c_0\)). In this way, \(|F_j|=|E_j|=v\), and thus we can exploit (3.6) with \(F=F_j\). A standard argument (see, e.g. [27, Theorem 21.11]) leads then to (3.7).
Now, since \(\partial \Omega \subset W\) is bounded, every connected component of \(\Omega \) with finite volume is bounded. Thus, by (a), (b) and (c) above, there is \(R>0\) such that \(W\cup E\,\subset \subset \,B_R\). Since \(|E\cap [B_{R+1}{\setminus } B_R]|=0\), we can pick \(T\in (R,R+1)\) such that \(\mathcal {H}^n(E_j\cap \partial B_T)\rightarrow 0\) and \(P(E_j{\setminus } B_T)=\mathcal {H}^n(E_j\cap \partial B_T)+P(E_j;\Omega {\setminus } B_T)\), and consider the sets \(F_j=(E_j\cap B_T)\cup B_{\rho _j}(y)\) corresponding to \(\rho _j=(|E_j\setminus B_T|/\omega _{n+1})^{1/(n+1)}\) and to \(y\in \mathbb {R}^{n+1}\) which is independent from j and such that \(|y|>\rho _j+T\) (notice that \(\sup _j\rho _j\le C(n)\,v^{1/(n+1)}\)). Since \(|F_j|=|E_j|=v\), (3.6) with \(F=F_j\) and \(P(B_{\rho _j})\le P(E_j\setminus B_T)\) give
so that, by the choice of T, \(\{F_j\}_j\) is a minimizing sequence for \(\psi _W(v)\), with \(F_j\subset B_{T^*}\) and \(T^*\) independent of j. We conclude by the Direct Method.
Step two: We prove (3.2). If \(E_v\) a minimizer of \(\psi _W(v)\) and \(R>0\) is such that \(W \subset \subset B_R\), then by \(P(E_v;\Omega )\le P(B^{(v)})\) we have, for \(v>v_0\), and \(v_0\) and \(C_0\) depending on n and W,
where we have used that, if \(v>2\,b>0\) and \(\alpha =n/(n+1)\), then
By combining (1.3) and (3.8) we conclude that, for some \(x\in \mathbb {R}^{n+1}\),
provided \(v>v_0\). Hence we deduce (3.2) from
Step three: We prove the existence of \(v_0\), \(\Lambda _0\), and \(s_0\) such that every \(E_v\in \textrm{Min}[\psi _W(v)]\) with \(v>v_0\) satisfies (3.1). Arguing by contradiction, we assume the existence of \(v_j\rightarrow \infty \), \(E_j\in \textrm{Min}[\psi _W(v_j)]\), \(F_j\subset \Omega \) with \(|F_j\Delta E_j|>0\) and \(F_j\Delta E_j\subset \subset B_{r_j}(x_j)\) for some \(x_j\in \mathbb {R}^{n+1}\) and \(r_j=v_j^{1/(n+1)}/j\), such that
Denoting by \(E_j^*\), \(F_j^*\) and \(\Omega _j\) the sets obtained by scaling \(E_j\), \(F_j\) and \(\Omega \) by a factor \(v_j^{-1/(n+1)}\), we find that \(F_j^*\Delta E_j^*\subset \subset B_{1/j}(y_j)\) for some \(y_j\in \mathbb {R}^{n+1}\), and
By (3.2) there are \(z_j\in \mathbb {R}^{n+1}\) such that \(|E_j^*\Delta B^{(1)}(z_j)|\rightarrow 0\). We can therefore use the volume-fixing variations of \(B^{(1)}\) to find diffeomorphisms \(\Phi ^j_t:\mathbb {R}^n\rightarrow \mathbb {R}^n\) and constants c(n) and C(n) such that, for every \(|t|<c(n)\), one has \(\{\Phi ^j_t\ne \textrm{id}\,\}\subset \subset U_j\) for some open ball \(U_j\) with \(U_j\subset \subset \Omega _j{\setminus } B_{1/j}(y_j)\), \(|\Phi ^j_t(E_j^*)\cap U_j|=|E_j^*\cap U_j|+t\), and \(P(\Phi ^j_t(E_j^*);U_j)\le (1+C(n)\,|t|)\,P(E_j^*;U_j)\). Since \(F_j^*\Delta E_j^*\subset \subset B_{1/j}(y_j)\) implies \(||F_j^*|-|E_j^*||<c(n)\) for j large, if \(t=|E_j^*|-|F_j^*|\), then \(G_j^*=\Phi ^j_t(F_j^*)\) is such that \(|G_j^*|=|E_j^*|\), and by \(E_j\in \textrm{Min}[\psi _W(v_j)]\),
Taking into account \(P(E^*_j;U_j)\le \psi _W(v_j)/v_j^{n/(n+1)}\le C(n)\), we thus find
which, by (3.9), gives \(j\,\big |E_j^*\Delta F_j^*\big |\le C(n)\,\big |E_j^*\Delta F_j^*\big |\). Since \(|E_j^*\Delta F_j^*|>0\), this is a contradiction for j large enough.
Step four: We now prove that, if \(\mathcal {R}(W)>0\), then
where x is related to \(E_v\) by (3.2). In proving (3.10) we will use the assumption \(\mathcal {R}(W)>0\) and the energy upper bound
A proof of (3.11) is given in step one of the proof of Theorem 1.6, see section 5; in turn, that proof is solely based on the results from section 4, where no part of Theorem 3.1 (not even the existence of minimizers in \(\psi _W(v)\)) is ever used. This said, when \(|W|>0\), and thus \(\mathcal {S}(W)>0\), one can replace (3.11) in the proof of (3.10) by the simpler upper bound
where, we recall, \(\mathcal {S}(W)=\sup \{\mathcal {H}^n(W\cap \Pi ):\Pi \text{ is } \text{ a } \text{ hyperplane } \text{ in } \mathbb {R}^{n+1}\}\). To prove (3.12), given \(\Pi \), we construct competitors for \(\psi _W(v)\) by intersecting \(\Omega \) with balls \(B^{(v')}(x_v)\) with \(v'>v\) and \(x_v\) such that \(|B^{(v')}(x_v){\setminus } W|=v\) and \(\mathcal {H}^n(W\cap \partial B^{(v')}(x_v))\rightarrow \mathcal {H}^n(W\cap \Pi )\) as \(v\rightarrow \infty \). Hence, \(\varlimsup _{v\rightarrow \infty }\psi _W(v)-P(B^{(v)})\le -\mathcal {H}^n(W\cap \Pi )\), thus giving (3.12). The proof of (3.11) is identical in spirit to that of (3.12), with the difference that to glue a large ball to \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\) we will need to establish the decay of \(\partial F\) towards a hyperplane parallel to \(\nu ^\perp \) to the high degree of precision expressed in (1.14). Now to prove (3.10): by contradiction, consider \(v_j\rightarrow \infty \), \(E_j\in \textrm{Min}[\psi _W(v_j)]\), and \(x_j\in \mathbb {R}^{n+1}\) with \(\inf _{x\in \mathbb {R}^{n+1}}|E_j\Delta B^{(v_j)}(x)|=|E_j\Delta B^{(v_j)}(x_j)|\), such that
and set \(\lambda _j=v_j^{-1/(n+1)}\), \(E_j^*=\lambda _j\,(E_j-x_j)\), \(W_j^*=\lambda _j\,(W-x_j)\), and \(\Omega _j^*=\lambda _j\,(\Omega -x_j)\). By (3.1), each \(E_j^*\) is a \((\Lambda _0,s_0)\)-perimeter minimizer with free boundary in \(\Omega _j^*\). By (3.2) and the defining property of \(x_j\), \(E_j^*\rightarrow B^{(1)}\) in \(L^1(\mathbb {R}^{n+1})\). Moreover, \(\textrm{diam}\,(W_j^*)\rightarrow 0\) and, by (3.13),
Thus there is \(z_0\not \in \partial B^{(1)}\) such that, for every \(\rho <\textrm{dist}(z_0,\partial B^{(1)})\), there is \(j(\rho )\) such that \(\{E_j^*\}_{j\ge j(\rho )}\) is a sequence of \((\Lambda _0,s_0)\)-perimeter minimizers in \(\mathbb {R}^{n+1}{\setminus } B_{\rho /2}(z_0)\). By Remark 3.2, up to increasing \(j(\rho )\), \((\partial E_j^*){\setminus } B_{\rho }(z_0)\) is contained in the normal graph over \(\partial B^{(1)}\) of \(u_j\) with \(\Vert u_j\Vert _{C^1(\partial B^{(1)})}\rightarrow 0\); in particular, by (3.14), \((\partial E_j^*)\setminus B_{\rho }(z_0)\) is disjoint from \(W_j^*\). By the constant mean curvature condition satisfied by \(\Omega \cap \partial E_j^*\), and by Alexandrov’s theorem [2], \((\partial E_j^*)\setminus B_{\rho }(z_0)\) is a sphere \(M_j^*\) for \(j\ge j(\rho )\). Let \(B_j^*\) be the ball bounded by \(M_j^*\). Since \(M_j^*\cap W_j^*=\emptyset \), we have either one of the following:
Case one: \(W_j^*\subset B_j^*\). We have \(\partial [B_j^*\cup E_j^*]\subset M_j^*\cup [(\partial E_j^*){\setminus } \textrm{cl}\,(B_j^*)]\subset (\partial E_j^*){\setminus } W_j^*\), so that, by \(|B_j^*\cup E_j^*|\ge |E_j^*|+|W_j^*|\ge 1\), we find \(P(E_j^*;\Omega _j^*)\ge P(B_j^*\cup E_j^*)\ge P(B^{(1)})\), that is, \(\psi _W(v_j)\ge P(B^{(1)})\), against (3.11).
Case two: \(W_j^*\cap B_j^*=\emptyset \). In this case, \(E_j^*=B_j^*\cup G_j^*\), where \(G_j^*\) is the union of the connected components of \(E_j^*\) whose boundaries have non-empty intersection with \(W_j^*\): in other words, we are claiming that \(B_j^*\) is the only connected component of \(E_j^*\) whose closure is disjoint from \(W_j^*\). Indeed, if this were not the case, we could recombine all the connected components of \(E_j^*\) with closure disjoint from \(W_j^*\) into a single ball of same total volume, centered far away from \(W_j^*\), in such a way to strictly decrease \(P(E_j^*;\Omega _j^*)\), against \(E_j\in \textrm{Min}[\psi _W(v_j)]\). Let us now set \(G_j=x_j+v_j^{1/(n+1)}\,G_j^*\) and \(U_j=x_j+v_j^{1/(n+1)}\,B_j^*\), so that \(E_j=G_j\cup U_j\) and \(\textrm{dist}(G_j,U_j)>0\).
If we start sliding \(U_j\) from infinity towards \(G_j\cup W\) along arbitrary directions, then at least one of the resulting “contact points” \(z_j\) belongs to \(\Omega \cap \partial G_j\): if this were not the case, then \(G_j\) would be contained in the convex envelope of W, so that \(|B_j|=|E_j|-|G_j|\ge v_j-C(W)\), and thus, by \(\psi _W(v_j)=P(E_j;\Omega )\ge P(B_j;W)=P(B_j)\), and by \(P(B_j)\ge P(B^{(v_j-C(W))})\ge P(B^{(v_j)})-C(W)\,v_j^{-1/(n+1)}\), against with (3.11) for j large.
By construction, there is a half-space \(H_j\) such that \(G_j\subset H_j\), \(z_j\in (\partial G_j)\cap (\partial H_j)\), and \(G_j\) is a perimeter minimizer in \(B_r(z_j)\) for some small \(r>0\). By the strong maximum principle, see, e.g. [13, Lemma 2.13], \(G_j\) has \(H_j-z_j\) as its unique blowup at \(z_j\). By De Giorgi’s regularity theorem, see e.g. [27, Theorem 21.8], \(G_j\) is an open set with smooth boundary in a neighborhood of \(z_j\). Therefore, if we denote by \(U_j'\) the translation of \(U_j\) constructed in the sliding argument, then, \(E_j'=G_j\cup U_j'\in \textrm{Min}[\psi _W(v)]\) and, in a neighborhood of \(z_j\), \(E_j'\) is the union of two disjoint sets with smooth boundary which touch tangentially at \(z_j\). In particular, \(|E_j'\cap B_r(z_j)|/|B_r|\rightarrow 1\) as \(r\rightarrow 0^+\), against volume density estimates implied by (3.1), see, e.g. [27, Theorem 21.11].
Step five: We finally show the existence of \(v_0\) and \(R_0(v)\) with \(R_0(v)\rightarrow 0^+\) and \(R_0(v)\,v^{1/(n+1)}\rightarrow \infty \), such that each \(E_v\in \textrm{Min}[\psi _W(v)]\) with \(v>v_0\) determines x and \(u\in C^{\infty }(\partial B^{(1)})\) such that (3.3) holds and \(\sup _{E_v} \Vert u \Vert _{C^1(\partial B^{(1)})}\rightarrow 0\) as \(v\rightarrow \infty \). To this end, let us consider \(v_j\rightarrow \infty \), \(E_j\in \textrm{Min}[\psi _W(v_j)]\), and define \(x_j\), \(E_j^*\) and \(W_j^*\) as in step four. Thanks to (3.10), there is \(z_0\in \partial B^{(1)}\) s.t. \(\textrm{dist}(z_0,W_j^*)\rightarrow 0\). In particular, for every \(\rho >0\), we can find \(j(\rho )\in \mathbb {N}\) such that if \(j\ge j(\rho )\), then \(E_j^*\) is a \((\Lambda _0,s_0)\)-perimeter minimizer in \(\mathbb {R}^{n+1}{\setminus } B_\rho (z_0)\), with \(E_j^*\rightarrow B^{(1)}\). By Remark 3.2, there are \(u_j\in C^1(\partial B^{(1)})\) such that
and \(\Vert u_j\Vert _{C^1(\partial B^{(1)})}\rightarrow 0\). By the arbitrariness of \(\rho \) and by a contradiction argument, (3.3) holds with \(R_0(v)\rightarrow 0^+\) such that \(R_0(v)\,v^{1/(n+1)}\rightarrow \infty \) as \(v\rightarrow \infty \), and with the uniform decay of \(\Vert u \Vert _{C^1(\partial B^{(1)})}\). \(\square \)
4 Properties of Isoperimetric Residues
Here we prove Theorem 1.1. It will be convenient to introduce some notation for cylinders and slabs in \(\mathbb {R}^{n+1}\): precisely, given \(r>0\), \(\nu \in \mathbb {S}^n\) and \(I\subset \mathbb {R}\), and setting \(\textbf{p}_{\nu ^\perp }(x)=x-(x\cdot \nu )\,\nu \) (\(x\in \mathbb {R}^{n+1}\)), we let
Given \(x\in \mathbb {R}^{n+1}\), we also set \({\textbf {D}}_r^\nu (x)=x+{\textbf {D}}_r^\nu \), \({\textbf {C}}_r^\nu (x)=x+{\textbf {C}}_r^\nu \), etc. We premise the following proposition, used in the proof of Theorem 1.1 and Theorem 1.6, and based on [32, Proposition 1 and Proposition 3].
Proposition 4.1
Let \(n\ge 2\), \(\nu \in \mathbb {S}^n\), and let f be a Lipschitz solution to the minimal surface equation on \(\nu ^\perp \setminus \textrm{cl}\,({\textbf {D }}_R^\nu )\). If \(n=2\), assume in addition that \(M=\{x+f(x)\,\nu :|x|>R\}\) is stable and has natural area growth, i.e.
Then there are \(a,b\in \mathbb {R}\) and \(c\in \nu ^\perp \) such that, for every \(|x|>R\),
Proof
If \(n\ge 3\), the fact that \(\nabla f\) is bounded allows one to represent f as the convolution with a singular kernel which, by a classical result of Littman, Stampacchia, and Weinberger [26], is comparable to the Green’s function of \(\mathbb {R}^n\); (4.4) is then deduced starting from that representation formula. For more details, see [32, Proposition 3]. In the case \(n=2\), by (4.2) and (4.3), we can exploit a classical “logarithmic cut-off argument” to see that M has finite total curvature, i.e. \(\int _M |K| \,d\mathcal {H}^2< \infty \), where K is the Gaussian curvature of M. As a consequence, see, e.g. [31, Section 1.2], the compactification \(\overline{M}\) of M is a Riemann surface with boundary, and M is conformally equivalent to \(\overline{M}{\setminus }\{p_1,...,p_m\}\), where \(p_i\) are interior points of \(\overline{M}\). One can thus conclude by the argument in [32, Proposition 1] that M has m-many ends satisfying the decay (4.5), and then that \(m=1\) thanks to the fact that \(M=\{x+f(x)\,\nu :|x|>R\}\).
Proof of Theorem 1.1
Step one: Given a hyperplane \(\Pi \) in \(\mathbb {R}^{n+1}\), if F is a half-space with \(\partial F=\Pi \) and \(\nu \) is a unit normal to \(\Pi \), then \(\textrm{res}_W(F,\nu )=\mathcal {H}^n(W\cap \Pi )\). Therefore the lower bound in (1.11) follows by
Step two: We notice that, if \((F,\nu )\in \mathcal {F}\), then by (1.8), (1.9), and the divergence theorem (see, e.g., [27, Lemma 22.11]), we can define a Radon measure on the open set \(\nu ^\perp \setminus \textbf{p}_{\nu ^\perp }(W)\) by setting
In particular, setting \(R'=\inf \{\rho : W\subset {\textbf {C }}_\rho ^\nu \}\), the fact that \(\mu ({\textbf {D }}_R^\nu \setminus \textbf{p}_{\nu ^\perp }(W))\ge 0\) gives
while the identity
(which possibly holds as \(-\infty =-\infty \) if \(P(F;{\textbf {C }}_{R'}^\nu \setminus W)=+\infty \)) gives that
In particular, the limsup defining \(\textrm{res}_W\) always exists as a limit.
Step three: We prove the existence of \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\) and (1.12). We first claim that if \(\{(F_j,\nu _j)\}_j\) is a maximizing sequence for \(\mathcal {R}(W)\), then, in addition to \(\textbf{p}_{\nu _j^\perp }(\partial F_j)=\nu _j^\perp \), one can modify \((F_j,\nu _j)\), preserving the optimality in the limit \(j\rightarrow \infty \), so that (writing \(X\subset ^{\mathcal {L}^{n+1}} Y\) for \(|X\setminus Y|=0\))
Indeed, since \((F_j,\nu _j)\in \mathcal {F}\), for some \(\alpha _j<\beta _j\in \mathbb {R}\) we have
Would it be that either \({\textbf {S}}_{(-\infty , \alpha _j)\cup (\beta _j,\infty )}^{\nu _j} \subset _{\mathcal {L}^{n+1}} F_j\) or \({\textbf {S}}_{(-\infty , \alpha _j)\cup (\beta _j,\infty )}^{\nu _j} \subset _{\mathcal {L}^{n+1}} \mathbb {R}^{n+1}{\setminus } F_j\), then, by the divergence theorem and by \(\textbf{p}_{\nu _j^\perp }(\partial F_j)=\nu _j^\perp \),
and thus \(\textrm{res}_{W}(F_j,\nu _j)=-\infty \); in particular, \((F_j,\nu _j)\in \mathcal {F}\) being a maximizing sequence, we would have \(\mathcal {R}(W)=-\infty \), against (4.6). This proves the validity (up to switching \(F_j\) with \(\mathbb {R}^{n+1}{\setminus } F_j\)), of the inclusions
Thanks to (4.9) (and by exploiting basic set operations on sets of finite perimeter, see, e.g., [27, Theorem 16.3]), we see that
in particular, \(\{(F_j^*,\nu _j)\}_j\) is also a maximizing sequence for \(\mathcal {R}(W)\). By standard compactness theorems there are F of locally finite perimeter in \(\mathbb {R}^{n+1}\) and \(\nu \in \mathbb {S}^n\) such that \(F_j\rightarrow F\) in \(L^1_{\textrm{loc}}(\mathbb {R}^{n+1})\) and \(\nu _j\rightarrow \nu \). If \(A\subset \subset {\textbf {C }}_R^\nu \setminus W\) is open, then, for j large enough, \(A\subset \subset {\textbf {C }}_R^{\nu _j}\setminus W\), and thus
By (4.7), \(R\mapsto \omega _n\,R^n-P(F_j;{\textbf {C }}_R^{\nu _j}{\setminus } W)\) is decreasing on \(R>R_j=\inf \{\rho :W\subset {\textbf {C }}_\rho ^{\nu _j}\}\). By \(\sup _jR_j\le C(W)<\infty \) and (4.11) we have
for every \(R>C(W)\); in particular, letting \(R\rightarrow \infty \),
By \(F_j\rightarrow F\) in \(L^1_{\textrm{loc}}(\mathbb {R}^{n+1})\), \(\partial F=\textrm{cl}\,(\partial ^*F)\) is contained in the set of accumulation points of sequences \(\{x_j\}_j\) with \(x_j\in \partial F_j\), so that (4.8) gives
if \([A,B]=\bigcap \{(\alpha ,\beta ):W\subset {\textbf {S}}^\nu _{(\alpha ,\beta )}\}\). Therefore \((F,\nu )\in \mathcal {F}\), and thus, by (4.12), \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\). We now show that (4.12) implies (1.12), i.e.
Indeed, should (4.14) fail, we could find \(\delta >0\) and \(G\subset \mathbb {R}^{n+1}\) with \(F\Delta G\subset \subset B\) for some ball B, such that \(P(G;B{\setminus } W)+\delta \le P(F;B{\setminus } W)\). For R large enough to entail \(B\subset \subset {\textbf {C }}_R^\nu \) we would then find
which, letting \(R\rightarrow \infty \), would violate the maximality of \((F,\nu )\) in \(\mathcal {R}(W)\).
Step four: We show that if \(\mathcal {R}(W)>0\) and \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\), then \(\partial F\subset \textbf{S}_{[A,B]}^\nu \) for A, B as in (4.13). Otherwise, by the same truncation procedure leading to (4.10) and by \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\), we would find
so that \((F^*,\nu )\in \textrm{Max}[\mathcal {R}(W)]\) too. Now \(P\big (F;{\textbf {C }}_R^{\nu _j}{\setminus } W\big ) - P\big (F^*;{\textbf {C }}_R^{\nu _j}{\setminus } W\big ) \) is increasing in R, and since \(\textrm{res}_W(F,\nu ) = \textrm{res}_W(F^*,\nu )\), it follows that \(P\big (F;{\textbf {C }}_R^{\nu _j}{\setminus } W\big ) = P\big (F^*;{\textbf {C }}_R^{\nu _j}{\setminus } W\big ) \) for large R. But this can hold only if \(\partial F \cap \Omega \) is an hyperplane disjoint from W, in which case \(\mathcal {R}(W)=\textrm{res}_W(F,\nu )=0\).
Step five: Still assuming \(\mathcal {R}(W)>0\), we complete the proof of statement (ii) by proving (1.14). By (4.13), if \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\), then \(F/R\rightarrow H^-=\{x\in \mathbb {R}^{n+1}:x\cdot \nu <0\}\) in \(L^1_{\textrm{loc}}(\mathbb {R}^{n+1})\) as \(R\rightarrow \infty \). By (4.14) and by improved convergence (i.e., Remark 3.2—notice carefully that \(\partial F\) is bounded in the direction \(\nu \) thanks to step four), we find \(R_F>0\) and functions \(\{f_R\}_{R>R_F}\subset C^1({\textbf {D }}_2^\nu {\setminus }{\textbf {D }}_1^\nu )\) such that
with \(\Vert f_R\Vert _{C^1({\textbf {D }}_2^\nu \setminus {\textbf {D }}_1^\nu )}\rightarrow 0\) as \(R\rightarrow \infty \). Scaling back to F we deduce that
for a (necessarily smooth) solution f to the minimal surfaces equation with
thanks to the fact that \(f(x)=R\,f_R(x/R)\) if \(x\in {\textbf {D }}_{2\,R}^\nu \setminus {\textbf {D }}_{R}^\nu \). When \(n\ge 3\), (1.14) follows by (4.15) and Proposition 4.1. When \(n=2\), (4.2) holds by (4.14). To check (4.3), we deduce by \(\textrm{res}_W(F,\nu )\ge 0\) the existence of \(R'>R_F\) such that \(\omega _n\,R^n\ge P(F;{\textbf {C }}_R^\nu {\setminus } W)-1\) if \(R>R'\). In particular, setting \(M=(\partial F)\setminus B_{R_F}\), for \(R>R'\) we have
provided \(C=\omega _n+[(1+\mathcal {H}^2(M\cap W))/(R')^n]\); while if \(R\in (R_F,R')\), then \(\mathcal {H}^2(M\cap B_R)\le C\,R^n\) with \(C=\mathcal {H}^2(M\cap B_{R'})/R_F^n\). This said, we can apply Proposition 4.1 to deduce (4.5). Since \(\partial F\) is contained in a slab, the logarithmic term in (4.5) must vanish (i.e. (4.5) holds with \(b=0\)), and thus (1.14) is proved. Finally, when \(n=1\), by (4.15) and (4.16) there are \(a_1,a_2\in \mathbb {R}\), \(x_1<x_2\), \(x_1,x_2\in \nu ^\perp \equiv \mathbb {R}\) such that \(f(x)=a_1\) for \(x\in \nu ^\perp \), \(x<x_1\), and \(f(x)=a_2\) for \(x\in \nu ^\perp \), \(x>x_2\). Now, setting \(M_1=\{x+a_1\,\nu :x\in \nu ^\perp ,x<x_1\}\) and \(M_2=\{x+a_2\,\nu :x\in \nu ^\perp ,x>x_2\}\), we have that
while, if L denotes the line through \(x_1+a_1\,\nu \) and \(x_2+a_2\,\nu \), then we can find \(\nu _L\in \mathbb {S}^1\) and a set \(F_L\) such that \((F_L,\nu _L)\in \mathcal {F}\) with \(\partial F_L=\big [\big ((\partial F){\setminus }(M_1\cup M_2)\big )\cup (L_1\cup L_2)\big ]\), where \(L_1\) and \(L_2\) are the two half-lines obtained by removing from L the segment joining \(x_1+a_1\,\nu \) and \(x_2+a_2\,\nu \). In this way, \(P(F_L;{\textbf {C }}_R^{\nu _L}{\setminus } W)=\mathcal {H}^n\big ({\textbf {C }}_R^\nu \cap (\partial F){\setminus }(W\cup M_1\cup M_2)\big )+2\,R-\big |(x_1+a_1\,\nu )-(x_2+a_2\,\nu )\big |\), so that \(\textrm{res}_W(F_L,\nu _L)-\textrm{res}_W(F,\nu )=\big |(x_1+a_1\,\nu )-(x_2+a_2\,\nu )\big |-|x_2-x_1|>0\), against \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\) if \(a_1\ne a_2\). Hence, \(a_1=a_2\).
We are left to prove that (4.15) holds with \(R_2=R_2(W)\) in place of \(R_F\), and the constants a, b, c and \(C_0\) appearing in (1.14) can be bounded in terms of W only. To this end, we notice that the argument presented in step one shows that \(\textrm{Max}[\mathcal {R}(W)]\) is pre-compact in \(L^1_\textrm{loc}(\mathbb {R}^{n+1})\). Using this fact and a contradiction argument based on improved convergence (Remark 3.2), we conclude the proof of statement (ii).
Step six: We complete the proof of statement (i) and begin the proof of statement (iii) by showing that, setting for brevity \(d=\textrm{diam}\,(W)\), it holds
whenever \(\Pi \) is a hyperplane in \(\mathbb {R}^{n+1}\). We have already proved the first inequality in step one. To prove the others, we notice that, if \((F,\nu )\in \mathcal {F}\), then \(\textbf{p}_{\nu ^\perp }(\partial F)=\nu ^\perp \) and (4.7)
give, for every \(R>R'\),
where in the last step we have used the isodiametric inequality. Maximizing over \((F,\nu )\) in (4.18) we complete the proof of (4.17). Moreover, if \(W=\textrm{cl}\,(B_{d/2})\), then, since \(\mathcal {S}(\textrm{cl}\,(B_{d/2}))=\mathcal {H}^n(\textrm{cl}\,(B_{d/2})\cap \Pi )=\omega _n\,(d/2)^n\) for any hyperplane \(\Pi \) through the origin, we find that \(\mathcal {R}(\textrm{cl}\,(B_{d/2}))=\omega _n\,(d/2)^n\); in particular, (4.17) implies (1.15).
Step seven: We continue the proof of statement (iii) by showing (1.16). Let \(\mathcal {R}(W)=\omega _n\,(d/2)^n\) and let \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\). Since every inequality in (4.18) holds as an equality, we find in particular that
By (4.20) and the discussion of the equality cases for the isodiametric inequality (see, e.g. [29]), we see that, for some \(x_0\in \nu ^\perp \),
Condition (4.19) implies that (1.14) holds with \(u\equiv a\) for some \(a\in [A,B]=\bigcap \{(\alpha ,\beta ):W\subset \textbf{S}^\nu _{(\alpha ,\beta )}\}\); in particular, since \((\partial F)\setminus W\) is a minimal surface and \(W\subset {\textbf {C }}_{d/2}^\nu (x_0)\), by analytic continuation we find that
By (4.21), we have that for \(R>R'\),
Going back to (4.18), this implies \(P(F;{\textbf {C }}_{d/2}^\nu (x_0)\setminus W)=0\). However, since \((\partial F){\setminus } W\) is (distributionally) a minimal surface, \(P(F;B_\rho (x)\setminus W)\ge \omega _n\,\rho ^n\) whenever \(x\in (\partial F){\setminus } W\) and \(\rho <\textrm{dist}(x,W)\), so that \(P(F;{\textbf {C }}_{d/2}^\nu (x_0)\setminus W)=0\) gives \(((\partial F){\setminus } W)\cap {\textbf {C }}_{d/2}^\nu (x_0)=\emptyset \). Hence, using also (4.21), we find \((\partial F){\setminus } W=\Pi {\setminus }\textrm{cl}\, \big (B_{d/2}(x)\big )\) for some \(x\in \Pi \), that is (1.16).
Step eight: We finally prove that \(\mathcal {R}(W)=\omega _n\,(d/2)^n\) if and only if there are a hyperplane \(\Pi \) and a point \(x\in \Pi \) such that
We first prove that the two conditions are sufficient. Let \(\nu \) be a unit normal to \(\Pi \) and let \(\Pi ^+\) and \(\Pi ^-\) be the two open half-spaces bounded by \(\Pi \). The condition \(\Pi \cup \partial B_{d/2}(x)\subset W\) implies \(W\subset {\textbf {C }}_{d/2}^\nu (x)\), and thus
In particular, \(\Omega \setminus (\Pi \setminus B_{d/2}(x))\) has a connected component F which contains
and since \(\Omega \setminus (\Pi \setminus B_{d/2}(x))\) contains exactly two unbounded connected components, it cannot be that F contains also \(\Pi ^-\setminus \textrm{cl}\,[{\textbf {C }}^\nu _{d/2,(-d,d)}(x)]\), therefore
As a consequence \(\partial F\) is contained in the slab \(\{y:|(y-x)\cdot \nu |<d\}\), and is such that \(\textbf{p}_{\nu ^\perp }(\partial F)=\nu ^\perp \), that is, \((F,\nu )\in \mathcal {F}\). Moreover, (4.24) implies
while the fact that F is a connected component of \(\Omega {\setminus } (\Pi {\setminus } B_{d/2}(x))\) implies \(\Omega \cap \partial F\subset \Pi {\setminus } \textrm{cl}\,(B_{d/2}(x))\). In conclusion, \(\Omega \cap \partial F=\Pi {\setminus }\textrm{cl}\,(B_{d/2}(x))\), hence
and \(\mathcal {R}(W)=\omega _n\,(d/2)^n\), as claimed. We prove that the two conditions are necessary. Let \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\). As proved in step seven, there is a hyperplane \(\Pi \) and \(x\in \Pi \) such that \(\Omega \cap \partial F=\Pi {\setminus } \textrm{cl}\,(B_{d/2}(x))\). If \(z\in \Pi \cap \partial B_{d/2}(x)\) but \(z\in \Omega \), then there is \(\rho >0\) such that \(B_\rho (z)\subset \Omega \), and since \(\partial F\) is a minimal surface in \(\Omega \), we would obtain that \(\Pi \cap B_\rho (z)\subset \Omega \cap \partial F\), against \(\Omega \cap \partial F=\Pi \setminus \textrm{cl}\,(B_{d/2}(x))\). So it must be \(\Pi \cap \partial B_{d/2}(x)\subset W\), and the necessity of (4.22) is proved. To prove the necessity of (4.23), we notice that since \(\Pi ^+\setminus \textrm{cl}\,[{\textbf {C }}^\nu _{d/2,(-d,d)}(x)]\) and \(\Pi ^-\setminus \textrm{cl}\,[{\textbf {C }}^\nu _{d/2,(-d,d)}(x)]\) are both open, connected, and unbounded subsets of \(\Omega {\setminus } (\Pi {\setminus } B_{d/2}(x))\), and since the complement in \(\Omega {\setminus } (\Pi {\setminus } B_{d/2}(x))\) of their union is bounded, it must be that \(\Omega \setminus (\Pi \setminus B_{d/2}(x))\) has at most two unbounded connected components: therefore we just need to exclude that it has only one. Assuming by contradiction that this is the case, we could then connect any point \(x^+\in \Pi ^+\setminus \textrm{cl}\,[{\textbf {C }}^\nu _{d/2,(-d,d)}(x)]\) to any point \(x^-\in \Pi ^-\setminus \textrm{cl}\,[{\textbf {C }}^\nu _{d/2,(-d,d)}(x)]\) with a continuous path \(\gamma \) entirely contained in \(\Omega {\setminus } (\Pi {\setminus } B_{d/2}(x))\). Now, recalling that \(\Omega \cap \partial F=\Pi {\setminus }\textrm{cl}\,(B_{d/2}(x))\), we can pick \(x_0\in \Pi {\setminus }\textrm{cl}\,(B_{d/2}(x))\) and \(r>0\) so that
and \(B_r(x_0)\cap \textrm{cl}\,[{\textbf {C }}^\nu _{d/2,(-d,d)}(x)]=\emptyset \). We can then pick \(x^+\in B_r(x_0)\cap \Pi ^+\), \(x^-\in B_r(x_0)\cap \Pi ^-\), and then connect them by a path \(\gamma \) entirely contained in \(\Omega {\setminus } (\Pi {\setminus } B_{d/2}(x))\). By (4.25), \(\gamma \) must intersect \(\partial F\), and since \(\gamma \) is contained in \(\Omega \), we see that \(\gamma \) must intersect \(\Omega \cap \partial F=\Pi \setminus \textrm{cl}\,(B_{d/2}(x))\), which of course contradicts the containment of \(\gamma \) in \(\Omega {\setminus } (\Pi {\setminus } B_{d/2}(x))\). We have thus proved that \(\Omega \setminus (\Pi \setminus B_{d/2}(x))\) has exactly two unbounded connected components. \(\square \)
5 Resolution Theorem for Exterior Isoperimetric Sets
The notation set in (4.1) is in use. Given \(v_j\rightarrow \infty \), we set \(\lambda _j=v_j^{1/(n+1)}\).
Proof of Theorem 1.6
Theorem 1.6–(i) and the estimate for \(|v^{-1/(n+1)}\,|x|-\omega _{n+1}^{-1/(n+1)}|\) in Theorem 1.6–(iv), have already been proved in Theorem 3.1–(ii), (iii).
Step one: We prove that
To this end, let \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\), so that by (1.13) and (1.14), we have
for a function \(f\in C^1(\nu ^\perp )\) satisfying
and for some \(a,b\in \mathbb {R}\) and \(c\in \nu ^\perp \) such that \(\max \{|a|,|b|,|c|\}\le C(W)<\infty \) (moreover, we can take \(b=0\), \(c=0\) and \(C_0=0\) if \(n=1\)). We are going to construct competitors for \(\psi _W(v)\) with v large by gluing a large sphere S to \(\partial F\) along \(\partial {\textbf {C }}_r^\nu \) for \(r>R_2\). This operation comes at the price of an area error located on the cylinder \(\partial {\textbf {C }}_r^\nu \). This error will remain bounded as needed thanks to the fact that (5.3) determines the distance (inside of \(\partial {\textbf {C }}_r^\nu \)) of \(\partial F\) from a hyperplane (namely, \(\partial G_r\) for the half-space \(G_r\) defined below) up to \(\textrm{o}(r^{1-n})\) as \(r\rightarrow \infty \). Thus, the asymptotic expansion (1.14) is just as precise as needed in order to perform this construction, i.e. our construction would not be possible with a less precise information.
The competitors \(F_{r,v}\) constructed in (5.7). A maximizer F in the isoperimetric residue \(\mathcal {R}(W)\) is joined to a ball of volume v, whose center \(x_{r,v}\) is determined by looking at best hyperplane \(\partial G_r\) approximating \(\partial F\) on the “lateral” cylinder \(\partial {\textbf {C }}_r^\nu \). To ensure the area error made in joining this large sphere to \(\partial F\) is negligible, the distance between \(\partial F\) and the sphere inside \(\partial {\textbf {C }}_r^\nu \) must be \(\textrm{o}(r^{1-n})\) as \(r\rightarrow \infty \). The asymptotic expansion (5.3) gives a hyperplane \(\partial G_r\) which is close to \(\partial F\) up to \(\textrm{O}(r^{-n})\), and is thus just as precise as needed to perform the construction
We now discuss the construction in detail. Given \(r>R_2\), we consider the half-space \(G_r\subset \mathbb {R}^{n+1}\) defined by the condition that
so that \(G_r\) is the “best half-space approximation” of F on \(\partial {\textbf {C }}_r^\nu \) according to (5.3). Denoting by \(\textrm{hd}\,(X,Y)\) the Hausdorff distance between \(X,Y\subset \mathbb {R}^{n+1}\), for every \(r>R_2\) and \(v>0\) we can define \(x_{r,v}\in \mathbb {R}^{n+1}\) in such a way that \(v\mapsto x_{r,v}\) is continuous and
Thus, the balls \(B^{(v)}(x_{r,v})\) have volume v and are locally converging in Hausdorff distance, as \(v\rightarrow \infty \), to the optimal half-space \(G_r\). Finally, we notice that by (5.3) we can find \(\alpha <\beta \) such that
and then define \(F_{r,v}\) by setting
see Fig. 5. We claim that, by using \(F_{r,v}\) as comparisons for \(\psi _W(|F_{r,v}|)\), and then sending first \(v\rightarrow \infty \) and then \(r\rightarrow \infty \), one obtains (5.1). We first notice that by (5.5) and (5.6) (see, e.g. [27, Theorem 16.16]), we have
where the last term is the “gluing error” generated by the mismatch between the boundaries of \(\partial F\) and \(\partial B^{(v)}(x_{r,v})\) along \(\partial _\ell {\textbf {C }}_{r,(\alpha ,\beta )}^\nu \). Now, thanks to (5.3) we have \(\textrm{hd}\,(G_r\cap \partial {\textbf {C }}_r^\nu ,F\cap \partial {\textbf {C }}_r^\nu )\le C_0\,r^{-n}\), so that
At the same time, by (5.5),
and thus we have the following estimate for the gluing error,
Again by (5.5), we find that
so that, by (5.11) and by the lower bound in (5.12), for every \(r>R_2\),
Combining (5.10) and (5.13) with (5.8) and the fact that \({\textbf {C }}_{r,(\alpha ,\beta )}^\nu \cap \partial F={\textbf {C }}_r^\nu \cap \partial F\) (see (5.6)), we find that, for every \(r>R_2\),
where (4.7) has been used. Now, combining the elementary estimates
with (5.14), we see that
Again by (5.15) and since \(v\mapsto |F_{r,v}|\) is a continuous function, we see that \(\varlimsup _{v\rightarrow \infty }\psi _W(|F_{r,v}|)-P(B^{(|F_{r,v}|)})=\varlimsup _{v\rightarrow \infty }\psi _W(v)-P(B^{(v)})\). This last identity combined with (5.16) implies (5.1) in the limit \(r\rightarrow \infty \).
Step two: Now let \(E_j\in \textrm{Min}[\psi _W(v_j)]\) for \(v_j \rightarrow \infty \). By (3.1) and a standard argument (see, e.g. [27, Theorem 21.14]), there is a local perimeter minimizer with free boundary F in \(\Omega \) such that, up to extracting subsequences,
Notice that it is not immediate to conclude from \(E_j\in \textrm{Min}[\psi _W(v_j)]\) that (for some \(\nu \in \mathbb {S}^n\)) \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\) (or even that \((\nu ,F)\in \mathcal {F}\)), nor that \(P(E_j;\Omega )-P(B^{(v_j)})\) is asymptotically bounded from below by \(-\textrm{res}_W(F,\nu )\). In this step we prove some preliminary properties of F, and in particular we exploit the blowdown result for exterior minimal surfaces contained in Theorem 2.1–(ii) to prove that F satisfies (5.2) and (5.3) (see statement (c) below). Then, in step three, we will use the decay rates (5.3) to show that \(E_j\) can be “glued” to F, similarly to the construction of step one, and then derive from the corresponding energy estimates the lower bound matching (5.1) and the optimality of F in \(\mathcal {R}(W)\).
(a) \(\Omega \cap \partial F\cap \partial B_\rho \ne \emptyset \) for every \(\rho \) such that \(W\subset \subset B_\rho \): If not there would be \(\varepsilon >0\) such that \(W\subset \subset B_{\rho -\varepsilon }\) and \(\Omega \cap \partial F\cap A_{\rho -\varepsilon }^{\rho +\varepsilon }=\emptyset \) (recall that \(A_r^s=\{x:s>|x|>r\}\)). By (5.17) and the constant mean curvature condition satisfied by \(\Omega \cap \partial E_j\), we would then find that each \(E_j\) (with j large enough) has a connected component of the form \(B^{(w_j)}(x_j)\), with \(B^{(w_j)}(x_j)\subset \subset \mathbb {R}^{n+1}{\setminus } B_{\rho +\varepsilon }\) and \(w_j\ge v_j-C(n)\,(\rho +\varepsilon )^{n+1}\). In particular, against \(\mathcal {R}(W)>0\),
(b) Sharp area bound: We combine the upper energy bound (5.1) with the perimeter inequality for spherical symmetrization, to prove
(Notice that (5.18) does not immediately imply the bound for \(P(F;\Omega \cap {\textbf {C }}_r^\nu )\) which would be needed to compare \(\mathcal {R}(W)\) and \(\textrm{res}_W(F,\nu )\).) To prove (5.18) we argue by contradiction, and consider the existence of \(\delta >0\) and r with \(W\subset \subset B_r\) such that \(P(F;\Omega \cap \ B_r)> \omega _n\,r^n-\mathcal {R}(W)+\delta \). In particular, for j large enough, we would then have
Again for j large, it must be \(\mathcal {H}^n(\partial E_j\cap \partial B_r)=0\): indeed, by (3.1), \(\Omega \cap \partial E_j\) has mean curvature of order \(\textrm{O}(\lambda _{j}^{-1})\), while of course \(\partial B_r\) has constant mean curvature equal to n/r. Thanks to \(\mathcal {H}^n(\partial E_j\cap \partial B_r)=0\),
If \(E_j^s\) denotes the spherical symmetral of \(E_j\) such that \(E_j^s\cap \partial B_\rho \) is a spherical cap in \(\partial B_\rho \), centered at \(\rho \,e_{n+1}\), with area equal to \(\mathcal {H}^n(E_j\cap \partial B_\rho )\), then we have the perimeter inequality
see [10]. Now, we can find a half-space J orthogonal to \(e_{n+1}\) and such that \(\mathcal {H}^n(J\cap \partial B_r)=\mathcal {H}^n(E_j\cap \partial B_r)\). In this way, using that \(|E_j^s\setminus B_r|=|E_j\setminus B_r|\) (by Fubini’s theorem in spherical coordinates), and that \(\mathcal {H}^n(B_r\cap \partial J)\le \omega _n\,r^n\) (by the fact that \(\partial J\) is a hyperplane), we find
which, with (5.19), (5.20) and (5.21), finally gives \(P(E_j;\Omega )-P(B^{(v_j)})> -\mathcal {R}(W)+\delta -C(n)\,r^{n+1}\,\lambda _{j}^{-1}\) for j large, against (5.1).
(c) Asymptotic behavior of \(\partial F\): We prove that there are \(\nu \in \mathbb {S}^n\), \(f\in C^\infty (\nu ^\perp )\), \(a,b\in \mathbb {R}\), \(c\in \nu ^\perp \), \(R'>\sup \{\rho :W\subset {\textbf {C }}_\rho ^\nu \}\) and C positive, with
To this end, by a standard argument exploiting the local perimeter minimality of F in \(\Omega \), given \(r_j\rightarrow \infty \), then, up to extracting subsequences, \(F/r_j{\mathop {\rightarrow }\limits ^{\textrm{loc}}}J\) in \(L^1_\textrm{loc}(\mathbb {R}^{n+1})\), where J is a perimeter minimizer in \(\mathbb {R}^{n+1}{\setminus }\{0\}\), \(0\in \partial J\) (thanks to property (a)), J is a cone with vertex at 0 (thanks to Theorem 2.7 and, in particular to (2.41)), and \(P(J;B_1)\le \omega _n\) (by (5.18)). If \(n\ge 2\), then \(\partial J\) has vanishing distributional mean curvature in \(\mathbb {R}^{n+1}\) (as points are removable singularities for the mean curvature operator when \(n\ge 2\)), thus \(P(J;B_1)\ge \omega _n\) by upper semicontinuity of area densities, and, finally, by \(P(J;B_1)=\omega _n\) and Allard’s regularity theorem, J is a half-space. If \(n=1\), then \(\partial J\) is the union of two half-lines \(\ell _1\) and \(\ell _2\) meeting at \(\{0\}\). If \(\ell _1\) and \(\ell _2\) are not opposite (i.e., if J is not a half-space), then we can find a half-space \(J^*\) such that \((J\cap J^*)\Delta J\subset \subset B\subset \subset \mathbb {R}^2{\setminus }\{0\}\) for some ball B, and \(P(J\cap J^*;B)<P(J;B)\), thus violating the fact that J is a perimeter minimizer in \(\mathbb {R}^{n+1}\setminus \{0\}\).
If \(n=1\) it is immediate from the above information that, for some \(R'>0\), \(F{\setminus } B_{R'}=J{\setminus } B_{R'}\); this proves (5.22) and (5.23) in the case \(n=1\). To prove (5.22) and (5.23) when \(n\ge 2\), we let \(M_0\) and \(\varepsilon _0\) be as in Theorem 2.1–(ii) with parameters n and \(\Gamma =2\,n\,\omega _n\), and with \(\sigma =1\). Since J is a half-space, by using Remark 3.2 and \(F/r_j{\mathop {\rightarrow }\limits ^{\textrm{loc}}}J\) on the annulus \(A_{1/2}^{2\,L}\), for some \(L>\max \{M_0,64\}\) to be chosen later on depending also on \(\varepsilon _0\), we find that
for \(f_j\in C^1(\nu ^\perp )\) with \(\Vert f_j\Vert _{C^1(\nu ^\perp )}\rightarrow 0\). By (5.24), \(V_j=\textbf{var}\,\big ((\partial F){\setminus } B_{r_j},1)\in \mathcal {V}_n(0,r_j,\infty )\), with (for \(\textrm{o}(1)\rightarrow 0\) as \(j\rightarrow \infty \))
By our choice of \(\Gamma \), by (5.18) and (5.25) we see that, for j large, we have
Moreover, we claim that setting
(so that, in particular, \(s_j>\max \{M_0,64\}\,r_j\)), then
provided j and L are taken large enough depending on \(\varepsilon _0\). To check the first inequality in (5.28) we notice that, by (5.25) and (5.26),
so that \(|\delta _{V_j,r_j,0}(s_j/8)|\le \varepsilon _0\) as soon as j is large with respect to \(\varepsilon _0\). Similarly, if \(r>s_j/8=(L\,r_j)/4\), then by (5.25), (5.26), (5.18), and \(r_j/r\le 4/L\),
provided j is large; hence the second inequality in (5.28) holds if L is large in terms of \(\varepsilon _0\). By (5.27) and (5.28), Theorem 2.1–(ii) can be applied to \((V,R,\Lambda ,s)=(V_j,r_j,0,s_j)\) with j large. As a consequence, passing from spherical graphs to cylindrical graphs with the aid of Lemma B.1, we find that, for some large j,
where \(f:\nu ^\perp \rightarrow \mathbb {R}\) is a smooth function which solves the minimal surfaces equation on \(\nu ^\perp \setminus B_{s_j/16}\). Since \(\partial F\) admits at least one sequential blowdown limit hyperplane (namely, \(\nu ^\perp =\partial J\)), by a theorem of Simon [36, Theorem 2] we find that \(\nabla f\) has a limit as \(|x|\rightarrow \infty \); in particular, \(|\nabla f|\) is bounded. Moreover, by (5.29) (or by the fact that F is a local perimeter minimizer in \(\Omega \)), \(\partial F\) is a stable minimal surface in \(\mathbb {R}^{n+1}\setminus B_{s_j/16}\), which, thanks to (5.18), satisfies an area growth bound like (4.3). We can thus apply Proposition 4.1 to deduce the validity of (5.23) when \(n\ge 3\), and of \(|f(x)-[a+b\,\log \,|x|+(c\cdot x)\,|x|^{-2}]|\le C\,|x|^{-2}\) for all \(|x|>R'\) when \(n=2\) (with \(R'>s_j\)). Recalling that F is a local perimeter minimizer with free boundary in \(\Omega \) (that is, \(P(F;\Omega \cap B)\le P(F';\Omega \cap B)\) whenever \(F\Delta F'\subset \subset B\subset \subset \mathbb {R}^3\)) it must be that \(b=0\), as it can be seen by comparing F with the set \(F'\) obtained by changing F inside \({\textbf {C }}_r^\nu \) (\(r>>R'\)) with the half-space \(G_r\) bounded by the plane \(\{x+t\,\nu :x\in \nu ^\perp ,t=a+b\,\log (r)+c\cdot x/r^2\}\) and such that \(\mathcal {H}^2((F\Delta G_r)\cap \partial {\textbf {C }}_r^\nu )\le C/r^2\) (we omit the details of this standard comparison argument). Having shown that \(b=0\), the proof of (5.23) when \(n=2\) also is complete and we are finished with (c).
(d) \(F\cup W\) defines an element of \(\mathcal {F}\): With \(R>R'\) as in (5.22) and (5.23), \(V_R=\textbf{var}\,((\partial F)\cap (B_R{\setminus } W))\) is a stationary varifold in \(\mathbb {R}^{n+1}{\setminus } K_R\) for \(K_R=W\cup \big \{x+f(x)\,\nu :x\in \nu ^\perp ,|x|=R\}\), and has bounded support. By the convex hull property [34, Theorem 19.2], we deduce that, for every \(R>R'\), \(\textrm{spt}V_R\) is contained in the convex hull of \(K_R\), for every \(R>R'\). Taking into account that \(f(x)\rightarrow a\) as \(|x|\rightarrow \infty \) we conclude that \(\Omega \cap \partial F\) is contained in the smallest slab \(\textbf{S}_{[\alpha ,\beta ]}^\nu \) containing both W and \(\{x:x\cdot \nu =a\}\). Now set \(F'=F\cup W\). Clearly \(F'\) is a set of locally finite perimeter in \(\Omega \) (since \(P(F';\Omega ')=P(F;\Omega ')\) for every \(\Omega '\subset \subset \Omega \)). Second, \(\partial F'\) is contained in \(\textbf{S}_{[\alpha ,\beta ]}^\nu \) (since \(\partial F'\subset [(\partial F)\cap \Omega ]\cup W\)). Third, by (5.22) and (5.23),
By combining (5.30) and (5.32) we see that \(\{x+t\,\nu :x\in \nu ^\perp ,t<\alpha \}\subset F'\), and by combining (5.31) and (5.32) we see that \(\{x+t\,\nu :x\in \nu ^\perp ,t>\beta \}\subset \mathbb {R}^{n+1}\setminus F'\): in particular, \(\textbf{p}_{\nu ^\perp }(\partial F')=\nu ^\perp \), and thus \((F',\nu )\in \mathcal {F}\).
Step three: We prove that
For \(v_j\rightarrow \infty \) achieving the liminf in (5.33), let \(E_j\in \textrm{Min}[\psi _W(v_j)]\) and let F be a (sub-sequential) limit of \(E_j\), so that properties (a), (b), (c) and (d) in step two hold for F. In particular, properties (5.22) and (5.23) from (c) are entirely analogous to properties (5.2) and (5.3) exploited in step one: therefore, the family of half-spaces \(\{G_r\}_{r>R'}\) defined by (5.4) is such that
(compare with (5.6), (5.9), and (5.12) in step one). By (5.35) we find
In order to relate the residue of \((F',\nu )\) to \(\psi _W(v_j)-P(B^{(v_j)})\) we consider the sets \(Z_j=(G_r\cap {\textbf {C }}_{r,(\alpha ,b)}^\nu )\cup (E_j\setminus {\textbf {C }}_{r,(\alpha ,\beta )}^\nu )\), which, by isoperimetry, satisfy
Since for a.e. \(r>R'\) we have
we conclude that
so that \(E_j\rightarrow F\) in \(L^1_\textrm{loc}(\mathbb {R}^{n+1})\) and (5.37) give, for a.e. \(r>R'\),
thanks to (5.34) and \((F\Delta G_r)\cap \partial {\textbf {C }}_r^\nu =(F\Delta G_r)\cap \partial {\textbf {C }}_{r,(\alpha ,\beta )}^\nu \). Letting \(r\rightarrow \infty \), recalling (5.36), and by \((F',\nu )\in \mathcal {F}\), we find \(\varliminf _{j\rightarrow \infty }\psi _W(v_j)-P(B^{(v_j)})\ge -\textrm{res}_W(F',\nu )\ge -\mathcal {R}(W)\). This completes the proof of (5.33), which in turn, combined with (5.1), gives (1.19), and also shows that \(L^1_\textrm{loc}\)-subsequential limits F of \(E_j\in \textrm{Min}[\psi _W(v_j)]\) for \(v_j\rightarrow \infty \) are such that, for some \(\nu \in \mathbb {S}^n\), \((F\cup W,\nu )\in \mathcal {F}\) and \(F'=F\cup W\in \textrm{Max}[\mathcal {R}(W)]\).
Step four: Moving towards the proof of (1.22), we prove the validity, uniformly among varifolds associated to maximizers of \(\mathcal {R}(W)\), of estimates analogous to (5.27) and (5.28). For a constant \(\Gamma >2\,n\,\omega _n\) to be determined later on (see (5.48), (5.49), and (5.50) below) in dependence of n and W, and for \(\sigma >0\), we let \(M_0=M_0(n,2\,\Gamma ,\sigma )\) and \(\varepsilon _0=\varepsilon _0(n,2\,\Gamma ,\sigma )\) be determined by Theorem 2.1. If \((F,\nu )\in {\textrm{Max}}[\mathcal {R}(W)]\), then by Theorem 1.1–(ii) we can find \(R_2=R_2(W)>0\), \(f\in C^\infty (\nu ^\perp )\) such that
and such that (1.14) holds with \(\max \{|a|,|b|,|c|\}\le C(W)\) and \(|\nabla f(x)|\le C_0/|x|^{n-1}\) for \(|x|>R_2\). Thus \(\Vert \nabla f\Vert _{C^0(\nu ^\perp {\setminus }{\textbf {D }}_r^\nu )}\rightarrow 0\) as \(r\rightarrow \infty \) uniformly on \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\), and there is \(R_3>\max \{2\,R_2,1\}\) (depending on W) such that, if \(V_F=\textbf{var}\,((\partial F)\setminus B_{R_3},1)\), then \(V_F\in \mathcal {V}_n(0,R_3,\infty )\), and
(compare with (5.27)). Then, arguing as in step three–(c), or more simply by exploiting (5.38) and the decay estimates (1.14), we see that there is \(L>\max \{M_0,64\}\), depending on n, W and \(\sigma \) only, such that, setting
we have for some \(c(n)>0\) (compare with (5.28))
Step five: Given \(E_j\in \textrm{Min}[\psi _W(v_j)]\) for \(v_j\rightarrow \infty \), we prove the existence of \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\) and \(h_j\in C^\infty ((\partial F){\setminus } B_{R_2})\) such that
and that if \(x_j\) satisfies \(|E_j\Delta B^{(v_j)}(x_j)|=\inf _x|E_j\Delta B^{(v_j)}(x)|\), then
finally, we prove statement (iii) (i.e., \((\partial E_j){\setminus } B_{R_2}\) is diffeomorphic to an n-dimensional disk). By step three, there is \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\) such that, up to extracting subsequences, (5.17) holds. By (5.17) and (5.38), and with \(s_W(\sigma )\) defined as in step four (see (5.40)) starting from F, we can apply Remark 3.2 to find \(f_j\in C^\infty (\nu ^\perp )\) such that
for j large enough (in terms of \(\sigma \), n, W, and F), and such that \(f_j\rightarrow f\) in \(C^1({\textbf {D }}_{s_W(\sigma )}^\nu \setminus {\textbf {D }}_{2\,R_2}^\nu )\). With \(R_3\) as in step four and with the goal of applying Theorem 2.1 to the varifolds \(V_j=\textbf{var}\,((\partial E_j){\setminus } B_{R_3},1)\), we notice that \(V_j\in \mathcal {V}_n(\Lambda _j,R_3,\infty )\), for some \(\Lambda _j\le \Lambda _0\,\lambda _{j}^{-1}\) (thanks to (3.1)). In particular, by (5.40), \(s_W(\sigma )\) satisfies the “mesoscale bounds” (compare with (2.2))
provided j is large. Moreover, by \(R_3>2\,R_2\) and \(s_W(\sigma )/8>2\,R_2\), by (5.38), (5.45) and \(f_j\rightarrow f\) in \(C^1\), we exploit (5.39) and (5.41) to deduce
We claim that, up to increasing \(\Gamma \) (depending on n and W), we can entail
Indeed, by Theorem 3.1–(i), for some positive \(\Lambda _0\) and \(s_0\) depending on W only, \(E_j\) is a \((\Lambda _0\,\lambda _{j}^{-1},s_0\,\lambda _{j})\)-perimeter minimizer with free boundary in \(\Omega \). Comparing \(E_j\) to \(E_j{\setminus } B_r\) by (3.1), for every \(r<s_0\,\lambda _{j}\),
since, at the same time, if \(r>s_0\,\lambda _{j}\), then
by combining (5.49) and (5.50) we find (5.48). With (5.47) and (5.48) at hand, we can also show that
Indeed, by \(s_W(\sigma )=2\,L\,R_3\) and by \(\Lambda _j\le \Lambda _0\,\lambda _{j}^{-1}\),
provided j is large enough. To complete checking that Theorem 2.1 can be applied to every \(V_j\) with j large enough, we now consider the quantities
and prove that, for a constant \(\tau _0\) depending on n and W only, we have
in particular, provided j is large enough, (5.52) implies immediately
which was the last assumption in Theorem 2.1 that needed to be checked. To prove (5.52), we pick \(\tau _0\) such that
(Of course this condition only requires \(\tau _0\) to depend on n; the dependence on W will appear later.) By definition of \(x_j\) and by (3.4), and up to extracting a subsequence, we have \(x_j\rightarrow z_0\) for some \(z_0\in \partial B^{(1)}\). In particular, setting \(\rho _j=\tau _0\,\lambda _{j}\), we find
thus proving that, for j large enough,
where we have used (5.47), \(\textrm{spt}\,\textrm{bd}_{V_j}\subset \partial B_{R_3}\), and (5.48). Therefore, provided we pick \(\tau _0\) depending on n and W so that \(C_{**}\,\tau _0\le \varepsilon _0/4\), and then we pick j large enough to entail \((C_*(n,W)/\tau _0^n)\lambda _{j}^{-1}\le \varepsilon _0/4\), we conclude that if \(r\in (R_3,\rho _j]\), then \(\delta _{V_j,R_3,\Lambda _j}(r)\ge \delta _{V_j,R_3,\Lambda _j}(\rho _j)\ge -\varepsilon _0\), where in the first inequality we have used Theorem 2.7–(i) and the fact that \(V_j\in \mathcal {V}_n(\Lambda _j,R_3,\infty )\). In summary, by (5.47) and (5.48) (which give (2.1)), by (5.46) (which gives (2.2) with \(s=s_W(\sigma )/8\)), and by (5.51) and (5.53) (which imply, respectively, (2.3) and (2.4)) we see that Theorem 2.1–(i) can be applied with \(V=V_j\) and \(s=s_W(\sigma )/8\) provided j is large in terms of \(\sigma \), n, W and the limit F of the \(E_j\)’s. Thus, setting
and noticing that by (5.52) and \(\Lambda _j\le \Lambda _0\,\lambda _{j}^{-1}\) we have
(for \(R_1\) depending on n and W only) we conclude that, for j large, there are \(K_j\in \mathcal {H}\) and \(u_j\in \mathcal {X}_\sigma (\Sigma _{K_j},\sigma _W(\sigma )/32,R_1\,\lambda _{j})\), such that
Similarly, by (5.39) and (5.41), thanks to Theorem 2.1–(ii) we have
for \(u\in \mathcal {X}_{\sigma '}(\Sigma _{\nu ^\perp },s_W(\sigma )/32,\infty )\) for every \(\sigma '>\sigma \). Now, by \(E_j\rightarrow F\) in \(L^1_\textrm{loc}(\mathbb {R}^{n+1})\), (5.54) and (5.55) can hold only if \(|\nu _{K_j}-\nu |\le \zeta (\sigma )\) for a function \(\zeta \), depending on n and W only, such that \(\zeta (\sigma )\rightarrow 0\) as \(\sigma \rightarrow 0^+\). In particular (denoting by \(\sigma _0^*\), \(\varepsilon _0^*\) and \(C_0^*\) the dimension dependent constants originally introduced in Lemma 2.5 as \(\sigma _0\), \(\varepsilon _0\) and \(C_0\)) we can find \(\sigma _1=\sigma _1(n,W)\le \sigma _0^*\) such that if \(\sigma <\sigma _1\), then \(\varepsilon _0^*\ge \zeta (\sigma )\ge |\nu _{K_j}-\nu |\), and correspondingly, Lemma 2.5–(i) can be used to infer the existence of \(u_j^*\in \mathcal {X}_{C_0\,(\sigma +\zeta (\sigma ))}(\Sigma _{\nu ^\perp },s_W(\sigma )/32,2\,R_1\,\lambda _{j})\) such that, for j large,
By (5.45) and Lemma B.1, (5.56) implies cylindrical graphicality: more precisely, provided \(\sigma _1\) is small enough, there are \(g_j\in C^1(\nu ^\perp )\) such that
At the same time, by (5.38), (1.14), and up to further increasing \(R_2\) and decreasing \(\sigma _1\), we can exploit Lemma B.2 in the Appendix to find \(h_j\in C^1(G(f))\), \(G(f)=\{x+f(x)\,\nu :x\in \nu ^\perp \}\), such that
which, combined with (5.38) and (5.58) shows that
that is (5.42). By \(E_j\rightarrow F\) in \(L^1_\textrm{loc}(\mathbb {R}^{n+1})\), we find \(h_j\rightarrow 0\) in \(L^1((\partial F)\cap A_{4\,R_2}^M)\) for every \(M<\infty \), so that, by elliptic regularity, (5.43) follows. We now recall that, by Theorem 3.1–(ii), \((\partial E_j){\setminus } B_{R_0(v_j)\,\lambda _{j}}\) coincides with
The overlapping of (5.58) and (5.59) (i.e., the fact that \(R_0(v_j)<R_1\) if j is large enough) implies statement (iii). Finally, combining (5.57) and (5.58) with (5.59) and \(\Vert w_j\Vert _{C^1(\partial B^{(1)})}\rightarrow 0\) we deduce the validity of (5.44). More precisely, rescaling by \(\lambda _j\) in (5.57) and (5.58) and setting \(E_j^*=E_j/\lambda _j\), we find \(g_j^*\in C^1(\nu ^\perp )\) such that, for every \(j\ge j_0(\sigma )\) and \(\sigma <\sigma _1\),
while rescaling by \(\lambda _j\) in (5.59) and setting \(z_j=x_j/\lambda _j\) we find
where \(||z_j|-\omega _{n+1}^{1/(n+1)}|\rightarrow 0\) thanks to (3.4). Up to subsequences, \(z_j\rightarrow z_0\), where \(|z_0|=\omega _{n+1}^{1/(n+1)}\). Should \(z_0\ne |z_0|\,\nu \), then picking \(\sigma \) small enough in terms of \(|\nu -(z_0/|z_0|)|>0\) and picking j large enough, we would then be able to exploit (5.60) to get a contradiction with \(\Vert w_j\Vert _{C^1(\partial B^{(1)})}\rightarrow 0\).
Conclusion: Theorem 3.1 implies Theorem 1.6–(i), and (1.19) was proved in step three. Should Theorem 1.6–(ii), (iii), or (iv) fail, then we could find a sequence \(\{(E_j,v_j)\}_j\) contradicting the conclusions of either step five or Theorem 3.1. We have thus completed the proof of Theorem 1.6. \(\square \)
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Appendices
Appendix A: Proof of Theorem 2.6
We assume \(H\in \mathcal {H}\), \(\Lambda \ge 0\), \(\eta _0>\eta >0\), \((r_1,r_2)\) and \((r_3,r_4)\) are \((\eta ,\eta _0)\)-related as in (2.36), and \(u\in \mathcal {X}_{\sigma }(\Sigma _H,r_1,r_2)\) is such that \(\Sigma _H(u,r_1,r_2)\) has mean curvature bounded by \(\Lambda \) in \(A_{r_1}^{r_2}\). We want to find \(\sigma _0\) and \(C_0\), depending on n, \(\eta _0\), and \(\eta \) only, such that, if \(\max \{1,\Lambda \,r_2\}\,\sigma \le \sigma _0\), then
and such that, if there is \(r\in (r_1,r_2)\) s.t. \(E_{\Sigma _H}^0[u_r]=0\) on \(\Sigma _H\), then
We make three preliminary considerations: (i): By [1, 4.5(8)]
Similarly, by the last displayed formula on [1, Page 236] and by [1, Lemma 4.9(1)], if \(\varphi =\psi ^2\,w\), \(w\in C^1(\Sigma _H\times (r_1,r_2))\) and \(\psi \in C^1(r_1,r_2)\), then
which is the second order expansion of the first variation of the area at \(\Sigma _H(u,r_1,r_2)\) along outer variations in spherical coordinates of the form \(\varphi =\psi ^2\,w\), \(\psi =\psi (r)\). (ii): For the sake of brevity, given \(\zeta :(r_1,r_2)\rightarrow \mathbb {R}\) a radial function, \(u,v:\Sigma _H\times (r_1,r_2)\rightarrow \mathbb {R}\), \(X,Y:\Sigma _H\times (r_1,r_2)\rightarrow \mathbb {R}^m\), we set
and \(Q_\zeta (u)=Q_\zeta (u,u)\), \(Q_\zeta (X)=Q_\zeta (X,X)\). (iii): The following two estimates (whose elementary proof is contained in [1, Lemma 7.13]) hold: whenever \(v\in C^1(\Sigma _H\times (r_1,r_2))\), we have
and, provided there is \(r\in [r_1,r_2]\) such that \(v_r=0\) on \(\Sigma _H\), we have
We are now ready for the proof. Compared to [1, Chapter 4], the main difference is that we replace [1, Lemma 4.10] with (A.7).
Step one: We prove that there is \(h:\Sigma _H\times (r_1,r_2)\rightarrow [-\Lambda ,\Lambda ]\) such that for every \(w\in C^1(\Sigma _H\times (r_1,r_2))\) and \(\psi \in C^1(r_1,r_2)\) we have
where \(T_\psi (u,w)=Q_\psi (\nabla ^{\Sigma _H} u,\nabla ^{\Sigma _H} w)+Q_{r}\big (\partial _ru,\partial _r[\psi ^2w]\big )-(n-1)\,Q_\psi (u,w)\). We start rewriting (A.4) as
If \(F_{u+t\,\varphi }:{\Sigma _H}\times (r_1,r_2)\rightarrow {\Sigma _H}(u+t\,\varphi ,r_1,r_2)\), \(\varphi =\psi ^2\,w\), is given by
then \(\{\Phi _t=F_{u+t\,\varphi }\circ (F_u)^{-1}\}_{t\in [0,1]}\) are diffeomorphisms on \({\Sigma _H}(u,r_1,r_2)\), with \(\Phi _t(\Sigma _H(u,r_1,r_2))=\Sigma _H(u+t\,\varphi ,r_1,r_2)\) and \(\dot{\Phi }_0=(d/dt)_{t=0}\Phi _t\). Since \(\Sigma _H(u,r_1,r_2)\) has mean curvature bounded by \(\Lambda \) in \(A_{r_1}^{r_2}\), for some bounded function \(h:\Sigma _H\times (r_1,r_2)\rightarrow [-\Lambda ,\Lambda ]\) we have
where \(\partial _i=\nabla _{\tau _i}\) for a local orthonormal frame \(\{\tau _i\}_{i=1}^{n-1}\) in \(\Sigma _H\), and where \(\star \) is the Hodge star-operator (so that \(\star \,(v_1\wedge v_2...\wedge v_n)\) is a normal vector to the hyperplane spanned by the \(v_i\)’s, with length equal to the n-dimensional volume of the parallelogram defined by the \(v_i\)’s, and whose orientation depends on the ordering of the \(v_i\)’s themselves). We can compute the initial velocity \(\dot{\Phi }_0\) of \(\{\Phi _t\}_{t\in [0,1]}\) by noticing that \(\Phi _t\big (F_u(\omega ,r)\big )=r\,(1+(u+t\,\varphi )^2)^{-1/2}\,(\omega +(u+t\,\varphi )\,\nu _H)\), so that,
At the same time
so that, with \(\Xi =\wedge _{i=1}^{n-1}\tau _i\), \(\hat{\tau }_i=\wedge _{j\ne i}\tau _j\), and \(P(u)^2=u^2+|\nabla ^{\Sigma _H} u|^2+(r\partial _r u)^2\),
for a coefficient \(G_i\) which we do not need to compute. Indeed, \(\star (\omega \wedge \nu _H\wedge \hat{\tau }_i)\), being parallel to \(\tau _i\), is orthogonal to \(\omega \) and \(\nu _H\), so that
In particular, since \(|h|\le \Lambda \),
Plugging this estimate into (A.8), and by \(\max \{1,\Lambda \,r_2\}\,\sigma \le \sigma _0\), we find (A.7).
Step two: We prove that
By \(Q_r(\psi \,\partial _ru,\psi 'u)\le Q_{r\,\psi }(\partial _ru)/4+C\,Q_{r\,\psi '}(u)\) and by (A.7)\(_{w=u}\) we find
which implies (A.9) provided \(\sigma _0\) is small enough.
Step three: We prove that, if \(w:\Sigma _H\times (r_1,r_2)\rightarrow \mathbb {R}\) is slice-wise orthogonal to \(u-w\), in the sense that \(\int _{\Sigma _H}w_r\,(u_r-w_r)=0\), \(\int _{\Sigma _H}\partial _rw_r\,(\partial _ru_r-\partial _rw_r)=0\), and \(\int _{\Sigma _H}\nabla ^{\Sigma _H} w_r\cdot (\nabla ^{\Sigma _H} u_r-\nabla ^{\Sigma _H} w_r)=0\) for every \(r\in (r_1,r_2)\), then
Indeed, by slice-wise orthogonality, we find that \(Q_\zeta (w)\le Q_\zeta (u)\), \(Q_\zeta (\partial _rw)\le Q_\zeta (\partial _ru)\) and \(Q_\zeta (\nabla ^{\Sigma _H} w)\le Q_\zeta (\nabla ^{\Sigma _H} u)\) whenever \(\zeta :(r_1,r_2)\rightarrow \mathbb {R}\) is radial. Therefore (A.7) gives \(|T_\psi (u,w)|\le Q_\psi (|w|,\Lambda \,r)+C(n)\,\sigma _0\,R_\psi (u)\), with \(R_\psi (u)=Q_\psi (u)+Q_\psi (\nabla ^{\Sigma _H} u)+Q_{r\,\psi }(\partial _ru)+Q_{r\,\psi '}(u)\). Combining this with (A.9) we get (A.10).
Step four: We prove (A.1). Let now \(\psi \) be a cut-off function between \((r_3,r_4)\) and \((r_1,r_2)\), so that with \(Z_\psi (u)=Q_\psi (\nabla ^{\Sigma _H}u)+Q_\psi (u)+Q_{r\,\psi }(\partial _ru)\),
If \(A(u)=\mathcal {H}^n(\Sigma _H(u,r_3,r_4))-\mathcal {H}^n(\Sigma _H(0,r_3,r_4))\), then by (A.3) with \((r_3,r_4)\) in place of \((r_1,r_2)\), we find
where in the last inequality we have used (A.9). We deduce
and (A.1) follows (with \(C_0=C_0(n,\eta _0,\eta )\) by the properties of \(\psi \)).
Step five: We finally prove that, if \(E^0_{\Sigma _H}[u_{r_*}]=0\) for some \(r_*\in (r_1,r_2)\), then (A.2) holds, that is
Define \(u^+,u^-,u^0:\Sigma _H\times (r_1,r_2)\rightarrow \mathbb {R}\) by setting, for \(r\in (r_1,r_2)\), \((u^+)_r=E^+_{\Sigma _H}[u_r]\), \((u^-)_r=E^-_{\Sigma _H}[u_r]\) and \((u^0)_r=E^0_{\Sigma _H}[u_r]\), where \(E^\pm _{\Sigma _H}\) denote the \(L^2(\Sigma _H)\)-orthogonal projections on the spaces of positive/negative eigenvectors of the Jacobi operator of \(\Sigma _H\), and where \(E^0_{\Sigma _H}\) is the \(L^2(\Sigma _H)\)-orthogonal projection onto the space of the Jacobi fields of \(\Sigma _H\). Since \((u^0)_{r_*}=0\), we can directly apply (A.6) with \(v=u^0\) and deduce that
By the orthogonality relations between \(u^0_r\), \(u^+_r\) and \(u^-_r\) we have that
By the spectral theorem, for every \(r\in (r_1,r_2)\) we have \( C_1(n)^{-1}\,\int _{\Sigma _H}(u^-)_r^2\le \,\int _{\Sigma _H}\,(n-1)\,(u^-)_r^2-|\nabla ^{\Sigma _H} (u^-)_r|^2\,, \) which, multiplied by \(r^{n-1}\,\psi ^2\), gives
where in the second to last identity we have used that \(w=u^-\) is slice-wise orthogonal to \(w-u\); in particular, by (A.10) with \(w=u^-\), we find
Again by slice-wise orthogonality of \(w=u^-\) to \(w-u\), we have
which combined into (A.15) gives
Using Hölder inequality again we have
Taking \(\psi \) to be a cut-off function between \((r_3,r_4)\) and \((r_1,r_2)\), we find
By combining (A.12), (A.16), and the analogous estimate to (A.16) for \(u^+\) with (A.13) and (A.14) we find that (A.16) holds with u in place of \(u^-\); this latter estimate, thanks to (A.5), finally gives (A.11).
Appendix B: Spherical and Cylindrical Graphs
We state here for the reader’s convenience two technical lemmas concerning spherical and cylindrical graphs. They are both used in the last step of the proof of Theorem 1.6. The elementary proofs are omitted.
Lemma B.1
(Spherical graphs as cylindrical graphs) There are dimension independent positive constants C and \(\eta _0\) with the following property. If \(n\ge 1\), \(H\in \mathcal {H}\) and \(u\in \mathcal {X}_\eta (\Sigma _H,r_1,r_2)\) with \(\eta <\eta _0\), then we have
and there is \(g\in C^1(H)\) such that \(\sup \big \{|x|^{-1}\,|g(x)|+|\nabla g(x)|:x\in H\big \}\le C\,\eta \) and \(\Sigma _H(u,r_1,r_2)=\big \{x+g(x)\,\nu _H:x\in \textbf{p}_H\big (\Sigma _H(u,r_1,r_2)\big )\big \}\). Moreover, if \((\rho _1,\rho _2)\subset ((1+C\,\eta )\,r_1,(1-C\,\eta ^2)\,r_2)\), then \(\Sigma _H(u,\rho _1,\rho _2)=\big \{x+g(x)\,\nu _H:x\in H\big \}\cap A_{\rho _1}^{\rho _2}\).
Lemma B.2
There is \(\eta \in (0,1)\) with the following property. If \(H\in \mathcal {H}\), \(R>1\), \(f\in C^2(H)\), and \(g\in C^1(H)\) are such that
then there is \(h\in C^1(G_H(f))\) such that
where \(G_H(f)=\{x+f(x)\,\nu _H:x\in H\}\) and, for \(z=x+f(x)\,\nu _H\), we have set \(\nu _f(z)=(1+|\nabla f(x)|^2)^{-1/2}\,(-\nabla f(x)+\nu _H)\).
Appendix C: Obstacles with Zero Isoperimetric Residue
Proposition C.1
If W is compact and \(\mathcal {R}(W)=0\), then \(\psi _W(v)-P(B^{(v)})\rightarrow 0\) as \(v\rightarrow \infty \) and W is purely \(\mathcal {H}^n\)-unrectifiable, in the sense that W cannot contain an \(\mathcal {H}^n\)-rectifiable set of \(\mathcal {H}^n\)-positive measure. In a partial converse, if W is purely \(\mathcal {H}^n\)-unrectifiable and \(\mathcal {H}^n(W)<\infty \), then \(\mathcal {R}(W)=0\).
Proof
Step one: Let \(\mathcal {R}(W)=0\). Comparing with balls, \(\varlimsup _{v\rightarrow \infty } \psi _W(v) - P(B^{(v)}) \le 0 = \mathcal {R}(W)\). To prove the matching lower bound, we argue by contradiction and consider \(E_j\in \textrm{Min}[\psi _W(v_j)]\) with \(v_j \rightarrow \infty \) such that
With (C.1) replacing \(\mathcal {R}(W)>0\), one can repeat verbatim step two-(a) of the proof of Theorem 1.1; we thus derive the asymptotic expansion for F as in step two-(c), which is then the key fact used in step three to derive that \(\lim _{j\rightarrow \infty } P(E_j;\Omega ) - P(B^{(v_j)}) \ge - {\textrm{res}}_W(F\cup W,\nu )\ge -\mathcal {R}(W)\); the latter inequality is of course in contradiction with (C.1) if \(\mathcal {R}(W)=0\). Next, arguing again by contradiction, we assume the existence of an \(\mathcal {H}^n\)-rectifiable set S with \(\mathcal {H}^n(W\cap S)>0\). By [34, Lemma 11.1], without loss of generality, S is a \(C^1\)-embedded hypersurface in \(\mathbb {R}^{n+1}\). Let x be a point of tangential differentiability for \(W\cap S\), so that \(\mathcal {H}^n(W\cap S\cap B_\rho (x))=\omega _n\,\rho ^n+\textrm{o}_x(\rho ^n)\) as \(\rho \rightarrow 0^+\). Since S is a \(C^1\)-embedded hypersurface, there is \(\nu \in \mathbb {S}^n\) such that for every \(\varepsilon >0\) there is \(\rho _*=\rho _*(x,\varepsilon )>0\) with \(S\cap {\textbf {C }}_{\rho _*,\rho _*}^\nu (x)=\{y+g(y)\,\nu :y\in {\textbf {D }}_{\rho _*}^\nu (x)\}\), where \(g\in C^1(x+\nu ^\perp )\) with \(g(x)=0\) and \(\textrm{Lip}(in)\le \varepsilon \). Denoting that \(G(g)=\{y+g(y)\,\nu :y\in (x+\nu ^\perp )\}\), and up to a decrease \(\rho _*\), we can get that
Since \(|g|\le \varepsilon \,\rho _*\) on \(\partial {\textbf {D }}_{\rho _*}^\nu (x)\), we can define \(f:(x+\nu ^\perp )\rightarrow \mathbb {R}\) so that \(f=g\) on \({\textbf {D }}_{\rho _*}^\nu (x)\), \(f=0\) on \((x+\nu ^\perp )\setminus {\textbf {D }}_{2\,\rho _*}^\nu (x)\), and \(\textrm{Lip}(f)\le \varepsilon \). Denoting by F the epigraph of f, we have that \((F,\nu )\in \mathcal {F}\) and we compute, for R large enough to entail that \({\textbf {C }}_{2\,\rho _*}^\nu (x)\cup W\subset \subset {\textbf {C }}_R^\nu \),
where we have used \(f=0\) on \(\nu ^\perp \setminus {\textbf {D }}_{2\,{\rho _*}}^\nu (x)\), (C.2) and \(\sqrt{1+\varepsilon ^2}\le 1+\varepsilon ^2\). Up to taking \(\varepsilon <\varepsilon (n)\), we thus find \(\textrm{res}_W(F,\nu )>0\), and thus deduce \(\mathcal {R}(W)>0\).
Step two: Let W be purely \(\mathcal {H}^n\)-unrectifiable with \(\mathcal {H}^n(W)<\infty \), and let \((F,\nu )\in \textrm{Max}[\mathcal {R}(W)]\). Since F is a local perimeter minimizer in \(\Omega \), F is open in \(\Omega \) with \(\Omega \cap \partial F=\textrm{cl}\,(\partial ^*F)\), where by \(\partial ^*F\) we mean the reduced boundary of F as a set of locally finite perimeter in \(\Omega \). Now, \(\omega _n\,R^n-P(F;{\textbf {C }}_R^\nu \setminus W)\) is decreasing towards \(\mathcal {R}(W)\ge \mathcal {S}(W)\ge 0\), therefore \(P(F;{\textbf {C }}_R^\nu {\setminus } W)<\infty \) for every R. In particular, \(\mathcal {H}^n\llcorner (\Omega \cap \partial F)\) is a Radon measure on \(\mathbb {R}^{n+1}\). Now, \(\partial F\subset (\Omega \cap \partial F)\cup W\), so that \(\mathcal {H}^n(W)<\infty \) implies that \(\mathcal {H}^n\llcorner \partial F\) is a Radon measure on \(\mathbb {R}^{n+1}\) and, since F is open, that F is a set of finite perimeter in \(\mathbb {R}^{n+1}\) by [17, Theorem 4.5.11]. The pure \(\mathcal {H}^n\)-unrectifiability of W gives \(P(F;{\textbf {C }}_R^\nu \setminus W)=P(F;{\textbf {C }}_R^\nu )\), where \(P(F;{\textbf {C }}_R^\nu )\ge \omega _n\,R^n\) by (1.8) and (1.9), and thus \(\mathcal {R}(W)=\textrm{res}_W(F,\nu )\le 0\). This proves \(\mathcal {R}(W)=0\). \(\square \)
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Maggi, F., Novack, M. Isoperimetric Residues and a Mesoscale Flatness Criterion for Hypersurfaces with Bounded Mean Curvature. Arch Rational Mech Anal 248, 87 (2024). https://doi.org/10.1007/s00205-024-02039-y
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DOI: https://doi.org/10.1007/s00205-024-02039-y