Equidistribution of minimal hypersurfaces for generic metrics

Abstract

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\le (n+1)\le 7$, we prove that there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in M. This gives a quantitative version of the main result of Irie et al. (Ann Math 187(3):963–972, 2018), that established density of minimal hypersurfaces for generic metrics. As in Irie et al. (2018), the main tool is the Weyl Law for the Volume Spectrum proven by Liokumovich et al. (Ann Math 187(3):933–961, 2018).

1 Introduction

In 1982, Yau [25] conjectured that every closed Riemannian three-manifold contains infinitely many smooth, closed, immersed minimal surfaces. In [8], Irie and the first two authors settled Yau’s conjecture in the generic case by proving a much stronger property holds true:

Theorem

(Irie, Marques, and Neves [8]) Let $M^{n+1}$ be a closed manifold of dimension $(n+1)$, with $3\le (n+1)\le 7$. Then for a $C^\infty$-generic Riemannian metric g on M, the union of all closed, smooth, embedded minimal hypersurfaces is dense.

In our paper, we use the methods of [8] in a more quantitative way and prove an even stronger property: there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in M.

Main Theorem

Let $M^{n+1}$ be a closed manifold of dimension $n+1$, with $3\le (n+1)\le 7$. Then for a $C^\infty$-generic Riemannian metric g on M, there exists a sequence $\{\Sigma _j\}_{j\in {\mathbb {N}}}$ of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in M: for any $f \in C^\infty (M)$ we have

$$\begin{aligned} \lim _{q \rightarrow \infty } \frac{1}{\sum _{j=1}^{q} \mathrm{vol}_g(\Sigma _{j})}\sum _{j=1}^{q} \int _{\Sigma _{j}} f \, d\Sigma _{j}=\frac{1}{\mathrm{vol}_g M}\int _M f dM. \end{aligned}$$

(1)

Even more, for any symmetric (0, 2)-tensor h on M we have:

$$\begin{aligned} \lim _{q \rightarrow \infty } \frac{1}{\sum _{j=1}^{q} \mathrm{vol}_g(\Sigma _{j})}\sum _{j=1}^{q} \int _{\Sigma _{j}} {{\,\mathrm{Tr}\,}}_{\Sigma _{j}} (h) \, d\Sigma _{j}=\frac{1}{\mathrm{vol}_g M}\int _M \frac{n{{\,\mathrm{Tr}\,}}_M h}{n+1} dM. \end{aligned}$$

(2)

Equidistribution theorems have an old history in fields like number theory, ergodic theory and harmonic analysis. Equidistribution of closed geodesics is known in some cases, like for compact hyperbolic manifolds (Bowen’ 72 [2] or Margulis [10], see also [26]). Equidistribution of totally geodesic surfaces is a well-studied problem for hyperbolic 3-manifolds [3, 14, 15, 17, 19]. Our theorem is the first of its kind for the higher-dimensional setting of minimal surfaces in general manifolds.

Remark

Yau’s Conjecture has been fully resolved by the third author [22]. He was able to localize the methods initially developed by the first two authors in [13], and proved that any compact $(M^{n+1},g)$, $3\le (n+1) \le 7$, contains infinitely many smooth, embedded, closed minimal hypersurfaces. It would be interesting to know whether density and equidistribution of minimal hypersurfaces hold for all Riemannian metrics.

As in the Irie–Marques–Neves paper [8], the crucial tool in our proof is the Weyl law for the Volume Spectrum conjectured by Gromov [5] and recently proven by the first two authors jointly with Liokumovich in [9]:

Weyl law for the volume spectrum (Liokumovich, Marques, and Neves [9]) There exists a universal constant$a(n)>0$such that for any compact Riemannian manifold$(M^{n+1},g)$we have:

$$\begin{aligned} \lim _{p\rightarrow \infty } \omega _p(M,g)p^{-\frac{1}{n+1}} = a(n) \mathrm{vol}(M,g)^\frac{n}{n+1}. \end{aligned}$$

The volume spectrum of a compact Riemannian manifold $(M^{n+1},g)$ is a nondecreasing sequence of numbers $\{\omega _p(M,g):p\in {\mathbb {N}}\}$ defined variationally by performing a min-max procedure for the area functional over multiparameter sweepouts. The first estimates for these numbers were proven in fundamental papers by Gromov in the late 1980s [4] and by Guth [6] more recently.

Our proof also uses a transversality argument, based on the Structure Theorem of White ([23], Theorem 2.1), that allows one to compute the derivative of the p-width as the derivative of the area of some minimal hypersurface. We combine this information with appropriately chosen N-parameter deformations of the metric, for N large, that generalize the one-parameter deformations of [7, 8]. A key idea in the paper (that can be be deduced from Lemma (2)) is that metrics which are critical points of the functional $g\mapsto \omega _p(M,g)p^{-\frac{1}{n+1}}-a(n) \mathrm{vol}(M,g)^\frac{n}{n+1}$ when restricted to the N-parameter family of deformations have minimal hypersurfaces that obey some form of equidistribution. The fact that this functional is only Lipschitz continuous and thus not differentiable everywhere is a serious technical issue that the authors had to overcome.

We note that Property (1) is equivalent to saying that

$$\begin{aligned} \frac{\sum _{j=1}^q \mu _{\Sigma _j}}{\sum _{j=1}^q \mu _{\Sigma _j}(M)} \rightarrow \frac{\mu }{\mu (M)} \end{aligned}$$

as measures, where $\mu _{\Sigma _j}=||\Sigma _j||$ is the Radon measure $\mu _{\Sigma _j}(U)=\mathrm{vol}_g(\Sigma _j \cap U)$, $U \subset M$, and $\mu =dv_g$ is the Riemannian volume measure of (M, g). Property (1) follows from Property (2) by choosing $h=f \cdot g$.

The dimensional restriction in the Main Theorem is due to the fact that in higher dimensions min–max (even area-minimizing) minimal hypersurfaces can have singular sets. We use Almgren–Pitts theory [1, 16], which together with Schoen–Simon regularity [18] produces smooth minimal hypersurfaces when $3\le (n+1) \le 7$. We expect that the methods of this paper can be generalized to handle the higher-dimensional singular case.

The Main Theorem raises many interesting and exciting new questions: namely whether equidistribution holds in the Grassmanian bundle, whether the minimal hypersurfaces realizing the width are the ones that become equidistributed, or whether there are conditions which ensure that the sequence of unit measures $\mu _{\Sigma _j}/\mu _{\Sigma _j}(M)$ converges to the normalized volume measure. Any progress related with these directions would be highly desirable.

2 Preliminaries

We suppose that M is a closed manifold of dimension $3 \le (n+1) \le 7$. For each $2\le q\le \infty$, we denote by $\Gamma _q$ the space of all $C^q$ Riemannian metrics on M, endowed with the $C^q$ topology. Given $g \in \Gamma _q$, we let ${\mathcal {V}}(g)$ be the set of stationary integral varifolds in (M, g) whose support is a closed, $C^2$, embedded, minimal hypersurface. Hence $V \in {\mathcal {V}}(g)$ if and only if there exist a disjoint collection $\{\Sigma _1,\ldots ,\Sigma _s\}$ of closed, $C^2$, embedded, connected minimal hypersurfaces in (M, g) and integers $\{m_1, \ldots , m_s\}\subset {\mathbb {N}}$ such that $V = m_1 \Sigma _1 + \cdots + m_s \Sigma _s$. By elliptic regularity, each $\Sigma _i$ is in fact of class $C^q$. The support of V is denoted by ${{\,\mathrm{spt}\,}}(V)$ and is equal to $\cup _{i=1}^s \Sigma _i$, while ||V|| denotes the Radon measure induced by V on M.

We denote by ${\mathcal {Z}}_n(M;\mathbb Z_2)$ the space of modulo two n-dimensional flat chains T in M with $T=\partial U$ for some $(n+1)$-dimensional modulo two flat chain U in M, endowed with the flat topology. This space is weakly homotopically equivalent to ${\mathbb {RP}}^\infty$ (see Section 4 of [12]). We denote by ${\overline{\lambda }}$ the generator of $H^1({\mathcal {Z}}_n(M;\mathbb Z_2), {\mathbb {Z}}_2)={\mathbb {Z}}_2$. The mass (n-dimensional volume) of T is denoted by M(T).

Let X be a finite dimensional simplicial complex. A continuous map $\Phi :X\rightarrow {\mathcal {Z}}_n(M;\mathbb Z_2)$ is called a p-sweepout if

$$\begin{aligned} \Phi ^*(\bar{\lambda }^p) \ne 0 \in H^p(X;\mathbb Z_2). \end{aligned}$$

We say X is p-admissible if there exists a p-sweepout $\Phi :X\rightarrow {\mathcal {Z}}_n(M;\mathbb Z_2)$ that has no concentration of mass, meaning

$$\begin{aligned} \lim _{r\rightarrow 0} \sup \{M(\Phi (x) \cap B_r(p)):x\in X, p\in M\}=0. \end{aligned}$$

The set of all p-sweepouts $\Phi$ that have no concentration of mass is denoted by $\mathcal P_p$. Note that two maps in $\mathcal P_p$ can have different domains.

In [13], the first two authors defined

Definition

The p-width of (M, g) is the number

$$\begin{aligned} \omega _p(M,g)=\inf _{\Phi \in \mathcal P_p}\sup \{M(\Phi (x)): x\in \mathrm{dmn}(\Phi )\}, \end{aligned}$$

where $\mathrm{dmn}(\Phi )$ is the domain of $\Phi$.

The next lemma gives that the normalized p-width $p^{-\frac{1}{(n+1)}}\omega _p(M,g)$ is a Lipschitz function of the metric on sets of uniformly equivalent metrics, with a Lipschitz constant that does not depend on p.

Lemma 1

Let ${\tilde{g}}$ be a $C^2$ Riemannian metric on M, and let $C_1<C_2$ be positive constants. Then there exists $K=K({\tilde{g}},C_1,C_2)>0$ such that

$$\begin{aligned} |p^{-\frac{1}{(n+1)}}\omega _p(M,g)-p^{-\frac{1}{(n+1)}}\omega _p(M,g')| \le K \cdot |g-g'|_{{\tilde{g}}} \end{aligned}$$

for any $g,g'\in \{h\in \Gamma _2 ; C_1{\tilde{g}} \le h \le C_2 {\tilde{g}}\}$ and any $p \in {\mathbb {N}}$.

Proof

It follows from the Gromov–Guth bound ([4, 6], see Theorem 5.1 of [13]) that there exists $C=C({\tilde{g}})$ such that $\omega _p(M, {\tilde{g}}) \le Cp^{\frac{1}{(n+1)}}$ for every $p\in {\mathbb {N}}$.

Given $g,g'\in \{h\in \Gamma _2 ; C_1{\tilde{g}} \le h \le C_2 {\tilde{g}}\}$, one can check (see Lemma 2.1 of [8]) that

$$\begin{aligned}&\omega _p(M,g')-\omega _p(M,g) \le \left( \left( \sup _{v \ne 0} \frac{g'(v,v)}{g(v,v)}\right) ^\frac{n}{2}-1\right) \omega _p(M,g)\\&\le \left( \left( 1 + \sup _{v \ne 0} \frac{|g(v,v)-g'(v,v)|}{g(v,v)}\right) ^\frac{n}{2}-1\right) \omega _p(M,g)\\&\le \left( \left( 1 +C_1^{-1} |g-g'|_{{\tilde{g}}}\right) ^\frac{n}{2}-1\right) \omega _p(M,g)\\&\le \left( \left( 1 +C_1^{-1} |g-g'|_{{\tilde{g}}}\right) ^\frac{n}{2}-1\right) C_2^\frac{n}{2}\omega _p(M,{\tilde{g}})\\&\le \left( \left( 1 +C_1^{-1} |g-g'|_{{\tilde{g}}}\right) ^\frac{n}{2}-1\right) C_2^\frac{n}{2}Cp^\frac{1}{n+1}, \end{aligned}$$

from which the result follows. $\square$

The next lemma concerns the differentiability properties of the p-width restricted to a generic finite-dimensional family of metrics. Let $I^N=[0,1]^N$.

Lemma 2

Let $g:I^N \rightarrow \Gamma _q$ be a smooth embedding, $N \in {\mathbb {N}}$. If $q \ge N+3$, then there exists an arbitrarily small perturbation in the $C^\infty$ topology $g':I^N \rightarrow \Gamma _q$ of g such that there is a subset ${\mathcal {A}}\subset I^N$ of full N-dimensional Lebesgue measure with the following property: for any $p\in {\mathbb {N}}$ and any point t of ${\mathcal {A}}$, the function $s \mapsto \omega _p(g'(s))$ is differentiable at t and there exists a disjoint collection $\{\Sigma _1, \ldots , \Sigma _Q\}$ of closed, $C^q$, embedded, minimal hypersurfaces of $(M,g'(t))$ together with integers $\{m_1, \ldots , m_Q\} \subset {\mathbb {N}}$ so that

$$\begin{aligned} \omega _p(g'(t)) = \sum _{j=1}^Q m_j \mathrm{vol}_{g'(t)}(\Sigma _j), \, \, \, \sum _{j=1}^Q \mathrm{index}(\Sigma _j) \le p, \end{aligned}$$

and

$$\begin{aligned} \frac{\partial }{\partial v}(\omega _p \circ g')_{|s=t}= & {} \frac{\partial }{\partial v} \left( \sum _{j=1}^Q m_j \mathrm{vol}_{g'(s)}(\Sigma _j)\right) _{|s=t}\\= & {} \sum _{j=1}^Q m_j \int _{\Sigma _j} \frac{1}{2} {{\,\mathrm{Tr}\,}}_{\Sigma _j, g'(t)} \left( \frac{\partial g'}{\partial v}_{|s=t}\right) d\Sigma _j \end{aligned}$$

for every $v \in {\mathbb {R}}^N$.

Proof

Let $g:I^N \rightarrow \Gamma _q$ be a smooth embedding. Consider a sequence $\{S_i\}_i$ that enumerates all the diffeomorphism types of n-dimensional closed manifolds, and let $\mathfrak {M}(S_i)$ be the Banach manifold of pairs $(\gamma , [u])$ as in the Structure Theorem of White [23] (Theorem 2.1), where $\gamma$ is a $C^q$ Riemannian metric and $u:S_i \rightarrow M$ is a $C^{2,\alpha }$ embedding that is minimal with respect to $\gamma$. Define $\mathfrak {M}:=\bigcup _i \mathfrak {M}(S_i)$ and the projection $\Pi :\mathfrak {M}\rightarrow \Gamma _q$ which sends $(\gamma ,[u])$ to $\gamma$. Theorem 2.1 of [23] (see also [24]) gives that $\mathfrak {M}$ is a separable $C^{q-2}$ Banach manifold and that $\Pi$ is a $C^{q-2}$ Fredholm map with Fredholm index zero. The pair $(\gamma , [u])$ is a critical point of $\Pi$ if and only if u admits a nontrivial Jacobi field with respect to the metric $\gamma$.

We can perturb $g:I^N \rightarrow \Gamma _q$ slightly in the $C^\infty$ topology to a $C^\infty$ embedding $g':I^N \rightarrow \Gamma _q$ that is transversal to $\Pi : \mathfrak {M} \rightarrow \Gamma _q$ by Smale’s Transversality Theorem (Theorem 3.1 of [21]). Transversality implies $\tilde{I}^N = \Pi ^{-1}(g'(I^N))$ is an N-dimensional submanifold of $\mathfrak {M}$ (Theorem 3.3 of [21]). Let $\pi =(g')^{-1} \circ \Pi _{|\tilde{I}^N}$, so $\pi : \tilde{I}^N \rightarrow I^N$. Let ${\mathcal {A}}'$ be the subset of points $t' \in I^N$ which are regular values of $\pi$ and such that the Lipschitz function $t \mapsto \omega _p(t):= \omega _p(M,g'(t))$ is differentiable at $t'$ for all p. This subset is of full Lebesgue measure in $I^N$ by Rademacher’s Theorem and Sard’s Theorem. Note that if $t' \in {\mathcal {A}}'$, then $g'(t')$ is a regular value of $\Pi$.

Consider

$$\begin{aligned}&{{\mathcal {S}}}'_{\kappa ,c}(p) := \Big \{t\in I^N; \exists V\in {\mathcal {V}}(g'(t)), ||V||_{g'(t)}(M)=\omega _p(t),\\&\qquad \qquad \qquad \mathrm{index}({{\,\mathrm{spt}\,}}(V))\le p, \max _{{{\,\mathrm{spt}\,}}(V)}|A| \le \kappa ,\\&\qquad \qquad \qquad \mathrm{\, and \,} V \mathrm{\, satisfies\,} (\star )_{c, \kappa , \sup _{s\in I^N}\omega _p(s)}\Big \} \end{aligned}$$

for any $\kappa >0$ and $c>0$, where

($\star _{c,\kappa , a}$):

: every two-sided, connected component of ${{\,\mathrm{spt}\,}}{V}$, has varifold distance (in the metric $g'(0)$) at least c of any varifold $2 \Sigma$, where $\Sigma$ is a one-sided, embedded, connected minimal hypersurface in $g'(s)$ with $|A|\le \kappa$ and $\mathrm{vol}_{g'(s)}(\Sigma ) \le a$ for some $s\in I^N$.

Each ${{\mathcal {S}}}'_{\kappa ,c}(p)$ is a closed set, by convergence properties of minimal hypersurfaces. Proposition 2.2 of [8] (which uses the index estimates of [11]) also holds for $C^q$ metrics if we allow the minimal hypersurfaces to be $C^2$. This follows, for instance, by approximating the $C^q$ metric by $C^\infty$ metrics, applying Proposition 2.2 of [8] to these metrics and using Sharp’s Compactness Theorem [20]. It implies

$$\begin{aligned} \cup _{\kappa , c \in {\mathbb {Q}}_+} \,{{\mathcal {S}}}'_{\kappa ,c}(p) = I^N \end{aligned}$$

for every $p \in {\mathbb {N}}$.

For $\kappa >0$ and $c>0$, define ${{\mathcal {S}}}_{\kappa ,c}(p)$ to be the set of points where the Lebesgue density of ${{\mathcal {S}}}'_{\kappa ,c}(p)$ is one. By the Lebesgue density theorem, ${{\mathcal {S}}}'_{\kappa ,c}(p)\backslash {{\mathcal {S}}}_{\kappa ,c}(p)$ has measure zero. Finally, define the full Lebesgue measure set

$$\begin{aligned} {\mathcal {A}}:={\mathcal {A}}'\cap \bigcap _p \bigcup _{\kappa , c}{{\mathcal {S}}}_{\kappa ,c}(p). \end{aligned}$$

Fix $p\in {\mathbb {N}}$, and let $t\in {\mathcal {A}}$. There exist $\kappa >0$ and $c>0$ such that $t\in {\mathcal {S}}_{\kappa ,c}(p)$. Since the Lebesgue density of ${\mathcal {S}}'_{\kappa ,c}(p)$ at t is one, we have that for any unit direction v, there is a sequence $\{t_m(v)\}_m\subset {\mathcal {S}}'_{\kappa ,c}(p)$ converging to t with $\frac{t_m(v)-t}{|t_m(v)-t|}$ converging to v, so that

$$\begin{aligned} \lim _{m\rightarrow \infty } \frac{\omega _p(t_m(v))-\omega _p(t)}{|t_m(v)-t|} = \frac{\partial }{\partial v} \omega _p(t). \end{aligned}$$

(3)

Fix v and a corresponding sequence $\{t_m(v)\}_m$. By construction, for each m there is a $V_m\in {\mathcal {V}}(g'(t_m(v)))$ with mass $\omega _p(t_m(v))$, with $\mathrm{index}({{\,\mathrm{spt}\,}}(V_m))\le p$, whose support has second fundamental form bounded by $\kappa$ (which is independent of m) and such that every two-sided, connected component of ${{\,\mathrm{spt}\,}}{V_m}$ has varifold distance (in the metric $g'(0)$) at least c (also independent of m) from any varifold $2 \Sigma$, where $\Sigma$ is a one-sided, connected minimal hypersurface in $g'(s)$ with $|A|\le \kappa$ and $\mathrm{vol}_{g'(s)}(\Sigma ) \le \sup _{s\in I^N}\omega _p(s)$ for some $s\in I^N$. This implies no two-sided component of ${{\,\mathrm{spt}\,}}{V_m}$ can collapse, after maybe passing to a subsequence, to a one-sided component with multiplicity two. Choosing a subsequence and renumbering if necessary, $V_m$ converges to a varifold $V\in {\mathcal {V}}(g'(t))$ and the supports ${{\,\mathrm{spt}\,}}(V_m)$ converge in $C^2$ to ${{\,\mathrm{spt}\,}}(V)$. This convergence is with multiplicity one, because if not one could construct by a standard argument a nontrivial Jacobi field on one of the components of ${{\,\mathrm{spt}\,}}(V)$. This is not possible, since $g'(t)$ is a regular value of $\Pi$.

Consider a sequence $\{\Sigma _m\}$ of connected components of ${{\,\mathrm{spt}\,}}(V_m)$ that converges in $C^2$ to $\Sigma$. By elliptic regularity, the convergence is also in $C^{2,\alpha }$. The corresponding points

$$\begin{aligned} {\tilde{z}}_m = (g'(t_m(v)), [\Sigma _m]) \in {\tilde{I}}^N \subset \mathfrak {M} \end{aligned}$$

converge to a point $z\in \Pi ^{-1}(g'(t)) \subset \tilde{I}^N$, $z=(g'(t),[\Sigma ])$. Note that since $t\in {\mathcal {A}}$, $\pi$ is a local diffeomorphism from a neighborhood of z in $\tilde{I}^N$ to a neighborhood of t in $I^N$. We write $\tilde{z}=(g'(\pi (\tilde{z})), [\Sigma (\pi (\tilde{z}))])$ for any $\tilde{z}$ in this neighborhood of z. For sufficiently large m, $[\Sigma _m]=[\Sigma (t_m(v))]$. But then

$$\begin{aligned}&\lim _{m\rightarrow \infty } \frac{\mathrm{vol}_{g'(t_m(v))}(\Sigma (t_m(v)))-\mathrm{vol}_{g'(t)}(\Sigma (t))}{|t_m(v)-t|} = \frac{\partial }{\partial v} \mathrm{vol}_{g'(s)}(\Sigma (s))_{|s=t}\\&=\frac{1}{2} \int _\Sigma {{\,\mathrm{Tr}\,}}_{\Sigma , g'(t)}\left( \frac{\partial g'}{\partial v}(t)\right) d \Sigma . \end{aligned}$$

Taking into account the multiplicity of each connected component of ${{\,\mathrm{spt}\,}}(V_m)$, the limit in (3) becomes

$$\begin{aligned} \frac{\partial }{\partial v} \omega _p(t)= \int _{V} \frac{1}{2} {{\,\mathrm{Tr}\,}}_{V, g'(t)} \left( \frac{\partial g'}{\partial v}_{|s=t}\right) d||V||(M), \end{aligned}$$

where V is of the form $\sum _{i=1}^Q m_i\tilde{\Sigma }_i$, with $\{\tilde{\Sigma }_1, \ldots , \tilde{\Sigma }_Q\}$ a disjoint collection of closed, $C^{2,\alpha }$, embedded, minimal hypersurfaces in $(M,g'(t))$ and $\{m_1, \ldots , m_Q\} \subset {\mathbb {N}}$, $||V||(M)=\omega _p(t)$, $\sum _{i=1}^Q \mathrm{index}(\tilde{\Sigma }_i) \le p$, $\max _{{{\,\mathrm{spt}\,}}(V)}|A| \le \kappa$ and V satisfies $(\star _{\kappa , c, \sup _{s\in I^N}\omega _p(s)})$. By elliptic regularity, each $\tilde{\Sigma }_i$ is of class $C^q$. Since t is a regular value of $\pi$, every embedded minimal hypersurface of $(M,g'(t))$ is non-degenerate. Because convergence of the supports can only happen with multiplicity one, there are only finitely many V’s as above, say $\{V^{(1)}, \ldots , V^{(P)}\}$. For any unit direction $v \in {\mathbb {R}}^N$, one has

$$\begin{aligned} \frac{\partial }{\partial v} \omega _p(t)= \int _{V^{(l)}} \frac{1}{2} {{\,\mathrm{Tr}\,}}_{V^{(l)}, g'(t)} \left( \frac{\partial g'}{\partial v}_{|s=t}\right) d||V^{(l)}||(M) \end{aligned}$$

for some $1\le l \le P$. This means that there will be a single $1\le l \le P$ such that the above formula is true for a linearly independent set $\{v_1,\ldots , v_N\}$, and hence for every v by linearity. This finishes the proof. $\square$

The next lemma concerns the gradient of Lipschitz functions that are almost constant. The convex hull of a set $K\subset \mathbb R^N$ is denoted by ${{\,\mathrm{Conv}\,}}(K)$.

Lemma 3

Given $\delta >0$ and $N \in {\mathbb {N}}$, there exists $\varepsilon >0$ depending on $\delta$ and N such that the following is true: for any Lipschitz function $f:I^N \rightarrow {\mathbb {R}}$ satisfying

$$\begin{aligned} |f(x)-f(y)|\le 2\varepsilon \end{aligned}$$

for every $x,y\in I^N$, and for any subset ${\mathcal {A}}$ of $I^N$ of full measure, there exist $N+1$ sequences of points $\{y_{1,m}\}_m,\ldots ,\{y_{N+1,m}\}_m$ contained in ${\mathcal {A}}$ and converging to a common limit $y\in (0,1)^N$ such that:

• f is differentiable at each $y_{i,m}$,

• the gradients $\nabla f(y_{i,m})$ converge to $N+1$ vectors $v_1,\ldots ,v_{N+1}$ with

  $$\begin{aligned} d_{{\mathbb {R}}^N}\left( 0, {{\,\mathrm{Conv}\,}}(v_1,\ldots ,v_{N+1})\right) < \delta , \end{aligned}$$

Proof

Suppose, by contradiction, that the lemma is false. Then there exists a sequence of Lipschitz functions $f_k:I^N \rightarrow {\mathbb {R}}$ satisfying

$$\begin{aligned} |f_k(x)-f_k(y)|\le 1/k \end{aligned}$$

for every $x,y\in I^N$, and a sequence of sets ${\mathcal {A}}_k \subset I^N$ of full measure, such that these sequences of points do not exist. Since $f_k$ is Lipschitz, the set $\mathcal {D}_k\subset I^N$ of points where $f_k$ is differentiable has full measure by Rademacher’s Theorem. Hence the set ${\mathcal {A}}'_k={\mathcal {A}}_k \cap \mathcal {D}_k$ has full measure also.

Choose a smooth function $g:I^N \rightarrow {\mathbb {R}}$ such that g is equal to 1 on the boundary of $I^N$ and equal to 0 at $(1/2,\ldots ,1/2)\in I^N$. Then the Lipschitz function $h_k=f_k-\frac{2}{k}g$ achieves its maximum at an interior point $y_k\in (0,1)^N$. Consider the set $V_k \subset {\mathbb {R}}^N$ of vectors v such that there exists a sequence $z_m \in {\mathcal {A}}'_k$ with $z_m \rightarrow y_k$ and $\nabla h_k(z_m)\rightarrow v$ as $m \rightarrow \infty$. The set $V_k$ is bounded and closed. For almost all directions w in the unit sphere $S^{N-1}$, the set

$$\begin{aligned} \{t\in [0,d_{{\mathbb {R}}^N}(y_k,\partial I^N)]: y_k+t w \in {\mathcal {A}}_k'\} \end{aligned}$$

has full measure in $[0,d_{{\mathbb {R}}^N}(y_k,\partial I^N)]$. For any such w, because $h_k$ has a maximum point at $y_k$, there exists $v \in V_k$ with $\langle v, w\rangle \le 0$. This implies that for any $w \in {\mathbb {R}}^N$, there exists $v\in V_k$ with $\langle v, w\rangle \le 0$. By the Hahn-Banach Theorem, $0 \in \mathrm{Conv}(V_k)$. Caratheodory’s Theorem gives vectors $\{\tilde{v}_1, \ldots , \tilde{v}_{N+1}\} \subset V_k$ such that $0 \in \mathrm{Conv}(\{\tilde{v}_1, \ldots , \tilde{v}_{N+1}\})$. Hence there exist $N+1$ sequences of points $\{y^{(k)}_{1,m}\}_m,\ldots ,\{y^{(k)}_{N+1,m}\}_m$ contained in ${\mathcal {A}}_k'$ and converging to $y_k\in (0,1)^N$ such that:

• $f_k$ is differentiable at each $y^{(k)}_{i,m}$,

• the gradients $\nabla f_k(y^{(k)}_{i,m})$ converge to $N+1$ vectors $v_1,\ldots ,v_{N+1}$ with

  $$\begin{aligned} d_{{\mathbb {R}}^N}\left( 0, {{\,\mathrm{Conv}\,}}(v_1,\ldots ,v_{N+1})\right) \le \frac{2}{k} \sup _{I^N} |\nabla g|. \end{aligned}$$

If k is sufficiently large, $\frac{2}{k} \sup _{I^N} |\nabla g| < \delta$. Contradiction. $\square$

The last lemma shows that one can make finitely many closed, embedded, minimal hypersurfaces nondegenerate by an arbitrarily small conformal change of the metric. This generalizes Proposition 2.3 of [8].

Lemma 4

Suppose $g\in \Gamma _q$, $q\ge 2$. Let $\{\Sigma _1, \ldots , \Sigma _L\}$ be a finite collection of closed, embedded, connected, $C^2$ minimal hypersurfaces in (M, g). Then there exists a sequence of metrics $g_i\in \Gamma _q$, $i\in {\mathbb {N}}$, converging to g in the $C^q$ topology such that $\Sigma _j$ is a nondegenerate minimal hypersurface in $(M,g_i)$ for all $j=1, \ldots , L$ and $i\in {\mathbb {N}}$.

Proof

Each $\Sigma _i$ is $C^q$ by elliptic regularity. We can suppose $\Sigma _j \ne \Sigma _k$ when $j \ne k$. Choose $\delta >0$ such that $B_r(q) \cap \Sigma _j$ is connected for every $j=1, \ldots , L$, $0<r\le \delta$ and $q \in \Sigma _j$. We claim that there exists a point $p \in \Sigma _1 \setminus (\Sigma _2 \cup \cdots \cup \Sigma _L)$.

Pick $x_1 \in \Sigma _1$ arbitrary. If $x_1 \notin \Sigma _2$, set $x_2=x_1$. Suppose $x_1 \in \Sigma _2$. If $B_\delta (x_1)\cap \Sigma _1 \subset \Sigma _2$, then $\Sigma _1=\Sigma _2$ by unique continuation. This is not possible, hence there exists $x_2 \in B_\delta (x_1)\cap \Sigma _1$ but $x_2 \notin \Sigma _2$. In any case, we have found $x_2 \in \Sigma _1 \setminus \Sigma _2$. Suppose we have $x_j \in \Sigma _1 \setminus (\Sigma _2 \cup \cdots \cup \Sigma _j)$, $2 \le j \le L-1$. If $x_j \notin \Sigma _{j+1}$, set $x_{j+1}=x_j$. Assume $x_j \in \Sigma _{j+1}$, and define $\delta _j = \min \{\delta , \frac{1}{2} d(x_j, \Sigma _2 \cup \cdots \cup \Sigma _j)\}>0.$ If $B_{\delta _j}(x_j) \cap \Sigma _1 \subset \Sigma _{j+1}$, then $\Sigma _1=\Sigma _{j+1}$ by unique continuation. This is impossible, hence there exists $x_{j+1} \in B_{\delta _j}(x_j) \cap \Sigma _1$ but $x_{j+1} \notin \Sigma _{j+1}$. In any case, we have found $x_{j+1} \in \Sigma _1 \setminus (\Sigma _2 \cup \cdots \cup \Sigma _{j+1})$. By induction, we find $x_L \in \Sigma _1 \setminus (\Sigma _2 \cup \cdots \cup \Sigma _L)$.

For similar reasons, there exist $p_l \in \Sigma _l \setminus (\cup _{k\ne l} \Sigma _k)$ for every $l=1, \ldots , L$. Choose $\eta >0$ sufficiently small so that $\eta$ is smaller than the injectivity radius of the manifold and such that $\eta <\frac{1}{4}d_g(p_l, \cup _{k\ne l} \Sigma _k)$ for every l. By decreasing $\eta$ if necessary, we can choose for each $l=1, \ldots , L$, a $C^q$ function $f_l:B_\eta (p_l) \rightarrow {\mathbb {R}}$ such that $f_l=0$ and $\langle \nabla f_l, N_l\rangle >0$ on $\Sigma _l \cap B_\eta (p_l)$, where $N_l$ is a local choice of unit normal to $\Sigma _l$. We also choose, for each $l=1, \ldots , L$, a smooth nonnegative function $\varphi _l:M \rightarrow {\mathbb {R}}$ such that $\varphi _l=1$ on $B_{\eta /2}(p_l)$ and $\varphi _l=0$ outside $B_{2\eta /3}(p_l)$.

Let $g_i = \exp (2\phi _i)g$, where $\phi _i=-\frac{1}{i}(\varphi _1f_1^2 + \cdots + \varphi _Lf_L^2)$. By the arguments of [8, Proposition 2.3], one can check that at any point y on $\Sigma _l$, $\phi _i=0$, $\nabla \phi _i=0$ and ${{\,\mathrm{Hess}\,}}_g\phi _i(N,N)= -\frac{2}{i}\varphi _l(y)\langle \nabla f_l, N\rangle ^2(y)$, where N is a unit normal to $\Sigma _l$ at y with respect to the metric g (or $g_i$). This implies $\Sigma _l$ remains minimal with respect to $g_i$ for every l, and at points of $\Sigma _l$ we have:

$$\begin{aligned} {{\,\mathrm{Ric}\,}}_{g_i}(N,N)+|A_{\Sigma _l,g_i}|^2_{g_i} = {{\,\mathrm{Ric}\,}}_{g}(N,N)+|A_{\Sigma _l,g}|^2_{g}+ \frac{2n}{i}\varphi _l\langle \nabla f_l, N\rangle ^2, \end{aligned}$$

where $|A_{\Sigma _l,g}|$ is the norm of the second fundamental form of $\Sigma _l$ with respect to g.

The Jacobi operator of $\Sigma _l$ acting on normal vector fields is given by

$$\begin{aligned} L_{\Sigma _l,g}(X) = \Delta ^\perp _{\Sigma _l,g}X+({{\,\mathrm{Ric}\,}}_g(N,N)+|A_{\Sigma _l,g}|^2_g)X. \end{aligned}$$

Since $g_i$ and g coincide on $\Sigma$, $\Delta ^\perp _{\Sigma _l,g}=\Delta ^\perp _{\Sigma _l,g_i}$ and hence

$$\begin{aligned} L_{\Sigma _l,g_i}(X)=L_{\Sigma _l,g}(X)+\frac{2n}{i}\varphi _l \langle \nabla f_l, N\rangle ^2X. \end{aligned}$$

(4)

Fix l, and define ${\tilde{L}}_t(X)=L_{\Sigma _l,g}(X) + t\varphi _l\langle \nabla f_l, N\rangle ^2X$ on $\Sigma _l$, for $t \in {\mathbb {R}}$. It is known that the eigenvalues of ${\tilde{L}}_t$ depend continuously on the parameter t. Suppose that $\Sigma _l$ is a degenerate minimal hypersurface in (M, g), and let Q be the unique integer such that $0=\lambda _Q({\tilde{L}}_0)<\lambda _{Q+1}({\tilde{L}}_0)$. If t is sufficiently small, then $\lambda _{Q+1}({\tilde{L}}_t)>0$.

Let X be in the zero eigenspace E of ${\tilde{L}}_0$, $X \ne 0$. Then

$$\begin{aligned}&\frac{d}{dt}_{|t=0}\left( -\frac{\int _{\Sigma _l} \langle {\tilde{L}}_t(X),X\rangle }{\int _{\Sigma _l}|X|^2}\right) \\&=\frac{d}{dt}_{|t=0}\left( \frac{-\int _{\Sigma _l}\langle {\tilde{L}}_0(X),X \rangle -t\int _{\Sigma _l} \varphi _l\langle \nabla f_l, N\rangle ^2 |X|^2}{\int _{\Sigma _l} |X|^2}\right) \\&= - \frac{\int _{\Sigma _l} \varphi _l\langle \nabla f_l, N\rangle ^2 |X|^2}{\int _{\Sigma _l}|X|^2} \\&\le - \frac{\int _{B_{\eta /2}(p_l) \cap \Sigma _l}\langle \nabla f_l, N\rangle ^2|X|^2}{\int _{\Sigma _l}|X|^2}. \end{aligned}$$

By unique continuation of solutions of linear elliptic equations and the finite-dimensionality of E, we can find a constant $c>0$ such that

$$\begin{aligned} \frac{d}{dt}_{|t=0}\left( -\frac{\int _{\Sigma _l} \langle {\tilde{L}}_t(X),X\rangle }{\int _{\Sigma _l}|X|^2}\right) \le -c \end{aligned}$$

(5)

for every $X \in E\setminus \{0\}$.

Recall the min-max characterization of the eigenvalue $\lambda _Q({\tilde{L}}_t)$:

$$\begin{aligned} \lambda _Q({\tilde{L}}_t) = \inf _W \max _{X\in W\backslash \{0\}} \frac{-\int _{\Sigma _l} \langle {\tilde{L}}_t(X),X \rangle }{\int _{\Sigma _{l}} |X|^2}, \end{aligned}$$

(6)

where the infimum is taken over all the Q-dimensional subspaces W of the space of smooth, normal vector fields on $\Sigma _l$. If $\tilde{W}$ is the subspace spanned by the eigensections of ${\tilde{L}}_0$ corresponding to eigenvalues $\lambda \le 0$, then $\mathrm{dim}(W)=Q$. By combining (5) and (6), we have

$$\begin{aligned} \lambda _Q({\tilde{L}}_t) \le \max _{X\in \tilde{W}\backslash \{0\}} \frac{-\int _{\Sigma _l} \langle {\tilde{L}}_t(X),X \rangle }{\int _{\Sigma _{l}} |X|^2}\le -\frac{c}{2}t \end{aligned}$$

for sufficiently small $t \ge 0$. Therefore for sufficiently large i we have both $\lambda _Q(L_{\Sigma _l,g_i})<0$ and $\lambda _{Q+1}(L_{\Sigma _l,g_i})>0$. This implies $\Sigma _l$ is nondegenerate with respect to $(M,g_i)$ for sufficiently large i. Since this is true for every $l=1, \ldots , L$, the Lemma is proved. $\square$

3 Proof of the main theorem

Let g be a smooth Riemannian metric on M, K be an integer and $\epsilon _1>0$ be a positive constant smaller than the injectivity radius of g. Let ${\hat{B}}_1,\ldots ,{\hat{B}}_K$ be disjoint domains in M, with piecewise smooth boundary, such that the union of their closures covers M.

Let $B_k$ be some neighborhood of ${\hat{B}}_k$. We suppose that each $B_k$ is contained in a geodesic ball of radius $\epsilon _1$. Choose also a smooth function $0\le \phi _k \le 1$ that is equal to 1 on ${\hat{B}}_k$ and with ${{\,\mathrm{spt}\,}}(\phi _k) \subset B_k$, and a point $q_k\in {\hat{B}}_k$ for each k. We can also suppose that $q_k \notin B_l$ if $l\ne k$. Define the partition of unity $\psi _k=\frac{\phi _k}{\sum _q\phi _q}$. Hence $\psi _k(q_k)=1$ and $\psi _k(q_l)=0$ for $l \ne k$.

For a fixed k, let e be a unit vector in the tangent space of M at $q_k$. It determines by parallel transport along geodesics starting at $q_k$ a unit vector field in $B_k$ still denoted by e. We define a nonnegative symmetric (0, 2)-tensor h(e) on $B_k$ as follows: $h(e)(v,w) = \langle v,e\rangle _g \langle w,e\rangle _g$.

Now consider the space ${\mathcal {B}}_k$ of orthonormal bases at $q_k$; these ${\mathcal {B}}_k$ are endowed with a natural metric determined by g and of course are isometric to each other. For each k, pick L points $x^k_1,\ldots ,x^k_L\in {\mathcal {B}}_k$ such that any point in ${\mathcal {B}}_k$ is at distance less than $\epsilon _1$ to one of the $x^k_l$. Each $x^k_l$ is an orthonormal basis $(x^k_{l,1},\ldots ,x^k_{l,n+1})$ at $q_k$ and so we can consider the family of symmetric (0, 2)-tensors $h^k_{l,j}=h(x^k_{l,j})$. Note that by construction, in $B_k$, for any l the sum $\sum _{j=1}^{n+1} h^k_{l,j}$ is the metric g.

We denote by ${\mathcal {C}}_{g, {\tilde{K}}, \epsilon _1}$ the set of all possible choices

$$\begin{aligned} (K, \{{\hat{B}}_k\}, \{B_k\}, \{\phi _k\}, \{q_k\}, \{x^k_l\}) \end{aligned}$$

as above, with $K \ge {\tilde{K}}$. The set ${\mathcal {C}}_{g, {\tilde{K}}, \epsilon _1}$ is non-empty, as can be seen by taking a sufficiently fine triangulation of M.

Recall that ${\mathcal {V}}(g)$ denotes the set of stationary integral varifolds in (M, g) whose support is an embedded minimal hypersurface. We claim that in order to show the main theorem, it suffices to prove the following property.

(P): for any metric g, for every $\epsilon _1>0$, ${\tilde{K}}>0$ and any choice of

$$\begin{aligned} S=(K, \{{\hat{B}}_k\}, \{B_k\}, \{\phi _k\}, \{q_k\}, \{x^k_l\}) \in {\mathcal {C}}_{g, {\tilde{K}}, \epsilon _1}, \end{aligned}$$

there is a metric ${\tilde{g}}$ arbitrarily close to g in the $C^\infty$ topology such that there are varifolds $V_1,\ldots ,V_J$ of ${\mathcal {V}}({\tilde{g}})$ whose support ${{\,\mathrm{spt}\,}}(V_j)$ are nondegenerate, and coefficients $\alpha _1,\ldots ,\alpha _J\in [0,1]$ with $\sum _i \alpha _i =1$ satisfying

$$\begin{aligned} \forall k,l,j \quad \Big | \sum _i \alpha _i \frac{V_i(\psi _k h^k_{l,j})}{||V_i||(M)} - \frac{1}{(n+1)}\frac{1}{\mathrm{vol}_{{\tilde{g}}}(M)}\int _M \psi _k dv_{{\tilde{g}}}\Big | < \epsilon _1/K, \end{aligned}$$

(7)

where the terms of the sum are computed for the metric ${\tilde{g}}$. Here

$$\begin{aligned} V(h)=\int _{G_n(M)} h(\nu ,\nu ) dV(p, \pi ), \end{aligned}$$

where $G_n(M)$ denotes the Grassmannian of n-dimensional planes of M and $\nu$ is a unit normal to the n-plane $\pi \subset T_pM$.

Indeed, let us explain why Property (P) implies the main theorem. We denote by $\mathcal {M}(g,\epsilon _1,{\tilde{K}}, S)$, with

$$\begin{aligned} S=(K, \{{\hat{B}}_k\}, \{B_k\}, \{\phi _k\}, \{q_k\}, \{x^k_l\})\in {\mathcal {C}}_{g, {\tilde{K}}, \epsilon _1}, \end{aligned}$$

the family of metrics ${\tilde{g}}\in \Gamma _\infty$ at distance less than $\epsilon _1/K$ to g (computed with respect to g) in the $C^K$ topology such that there are $\{V_1,\ldots ,V_J\} \subset {\mathcal {V}}({\tilde{g}})$ whose supports are nondegenerate, $\alpha _1,\ldots ,\alpha _J\in [0,1]$ with $\sum _i \alpha _i =1$, which satisfy (7) for all k, l, j. If $g' \in \mathcal {M}(g,\epsilon _1,{\tilde{K}}, S)$, and $\{\Sigma _1', \ldots , \Sigma _Q'\}$ is any finite collection of nondegenerate minimal hypersurfaces in $(M,g')$, then for every metric ${\tilde{g}}$ that is sufficiently close to $g'$, there is a unique collection $\{\tilde{\Sigma }_1, \ldots , \tilde{\Sigma }_Q\}$ of nondegenerate minimal hypersurfaces in $(M,{\tilde{g}})$ such that $\tilde{\Sigma }_i$ is close to $\Sigma _i'$, $i=1, \ldots , Q$. Moreover, $\tilde{\Sigma }_i$ converges smoothly to $\Sigma _i'$ as ${\tilde{g}}$ converges to $g'$. This implies that $\mathcal {M}(g,\epsilon _1,{\tilde{K}}, S)$ is open in the $C^\infty$ topology.

Define

$$\begin{aligned} \mathcal {M}(\epsilon _1,{\tilde{K}}) : = \bigcup _{g\in \Gamma _\infty }\bigcup _{S \in {\mathcal {C}}_{g, {\tilde{K}}, \epsilon _1}} \mathcal {M}(g,\epsilon _1,{\tilde{K}},S). \end{aligned}$$

It is clearly open. Given an arbitrary metric $g \in \Gamma _\infty$, we can choose $S \in {\mathcal {C}}_{g, {\tilde{K}}, \epsilon _1}$. Property (P) implies that the metric g is a limit of metrics in $\mathcal {M}(g,\epsilon _1,{\tilde{K}},S)$. This shows that $\mathcal {M}(\epsilon _1,{\tilde{K}})$ is also dense.

Define

$$\begin{aligned} \mathcal {M}:= \bigcap _{m\in {\mathbb {N}}} \mathcal {M}(1/m,m). \end{aligned}$$

Since each $\mathcal {M}(1/m,m)$ is open and dense, the intersection $\mathcal {M}$ is a residual subset (in the Baire sense) of the set of metrics. We will show that for any metric in $\mathcal {M}$, one can find sequences of minimal hypersurfaces like in the Main Theorem. For any metric, a symmetric (0, 2)-tensor h is diagonalizable at every point. The idea is to find a fine subdivision of M in domains $B_k$ where h is approximately diagonal when expressed in the basis $x^k_{l(k)}$ for a certain $l(k)\in \{1,\ldots ,L\}$.

Let ${\tilde{g}} \in \mathcal {M}$. Then ${\tilde{g}}\in \mathcal {M}(1/m,m)$ for every $m \in {\mathbb {N}}$. Fix m. Then by construction there exists a metric g such that ${\tilde{g}} \in \mathcal {M}(g,1/m,m,S)$ for some choice of

$$\begin{aligned} S=(K, \{{\hat{B}}_k\}, \{B_k\}, \{\phi _k\}, \{q_k\}, \{x^k_l\}) \in {\mathcal {C}}_{g, m, 1/m}. \end{aligned}$$

In particular, g belongs to a 1 / (mK)-neighborhood of ${\tilde{g}}$ in the $C^K$ topology. We also have $\{V_1,\ldots ,V_J\} \subset {\mathcal {V}}({\tilde{g}})$, $\alpha _1,\ldots ,\alpha _J\in [0,1]$ with $\sum _i \alpha _i =1$, which satisfy

$$\begin{aligned} \forall k,l,j \quad \Big | \sum _i \alpha _i \frac{V_i(\psi _k h^k_{l,j})}{||V_i||(M)} - \frac{1}{(n+1)}\frac{1}{\mathrm{vol}_{{\tilde{g}}}(M)}\int _M \psi _k dv_{{\tilde{g}}}\Big | < 1/(mK). \end{aligned}$$

(8)

Note that $g, S, J, \{V_j\}, \{\alpha _j\}$ all depend on m.

Let h be a symmetric (0, 2)-tensor on M. The following computations are done with respect to the metric ${\tilde{g}}$, unless otherwise specified. We start by writing

$$\begin{aligned} \int _M{{\,\mathrm{Tr}\,}}(h) = \sum _k \int _{B_k} \psi _k {{\,\mathrm{Tr}\,}}(h), \end{aligned}$$

and

$$\begin{aligned} \sum _i \alpha _i \frac{V_i(h)}{||V_i||(M)} = \sum _k\sum _i \alpha _i \frac{V_i(\psi _k h)}{||V_i||(M)}. \end{aligned}$$

At each $q_k\in {\hat{B}}_k$, h is diagonalizable for the metric g in a g-orthonormal basis

$$\begin{aligned} u^k=(u^k_1,\ldots ,u^k_{n+1})\in {\mathcal {B}}_k \end{aligned}$$

with eigenvalues $\lambda _1(k),\ldots ,\lambda _{n+1}(k)$ and we note that $\sum _j \lambda _j(k)$ is the trace of h at $q_k$ for the metric g. Let l(k) be such that $x^k_{l(k)}$ is at distance less than 1 / m from $u^k$ in ${\mathcal {B}}_k$. We get on $B_k$ (which are contained in balls of radius 1 / m) the following estimates with the metric ${\tilde{g}}$:

$$\begin{aligned}&\displaystyle \left| h - \sum _{j=1}^{n+1} \lambda _j(k) h(u^k_j)\right| _{{\tilde{g}}} < \frac{C}{m}, \end{aligned}$$

(9)

$$\begin{aligned}&\left| \sum _{j=1}^{n+1} \lambda _j(k) h(u^k_j) - \sum _{j=1}^{n+1} \lambda _j(k) h^k_{l(k),j}\right| _{{\tilde{g}}} < \frac{C}{m}. \end{aligned}$$

(10)

Here C depends only on ${\tilde{g}}$ and h, and might be different from line to line.

We have, by (8), that

$$\begin{aligned}&\forall k \quad \Big | \sum _i \sum _j \alpha _i \frac{V_i(\psi _k \lambda _j(k)h^k_{l(k),j})}{||V_i||(M)} - \frac{1}{(n+1)}\frac{1}{\mathrm{vol}(M)}\int _M (\sum _j\lambda _j(k)\psi _k )\Big | \\&\quad < C/(mK). \end{aligned}$$

Hence

$$\begin{aligned}&\sum _k \Big | \sum _i \sum _j \alpha _i \frac{V_i(\psi _k \lambda _j(k)h^k_{l(k),j})}{||V_i||(M)} - \frac{1}{(n+1)}\frac{1}{\mathrm{vol}(M)}\int _M (\sum _j\lambda _j(k)\psi _k) \Big | \\&\quad < C/m, \end{aligned}$$

and since $|{{\,\mathrm{Tr}\,}}_g h - {{\,\mathrm{Tr}\,}}_{{\tilde{g}}} h| <C/m$, we obtain readily

$$\begin{aligned}&\sum _k \Big | \sum _i \sum _j \alpha _i \frac{V_i(\psi _k \lambda _j(k)h^k_{l(k),j})}{||V_i||(M)} - \frac{1}{(n+1)}\frac{1}{\mathrm{vol}(M)}\int _M \psi _k {{\,\mathrm{Tr}\,}}h\Big | < C/m. \end{aligned}$$

Therefore

$$\begin{aligned} \Big | \sum _k \sum _i \sum _j \alpha _i \frac{V_i(\psi _k \lambda _j(k)h^k_{l(k),j})}{||V_i||(M)} - \frac{\int _{M}{{\,\mathrm{Tr}\,}}h}{(n+1)\mathrm{vol}(M)}\Big | < C/m. \end{aligned}$$

But we also have by (9) and (10) that

$$\begin{aligned} \Big | \sum _k\sum _i \sum _j \alpha _i \frac{V_i(\psi _k \lambda _j(k)h^k_{l(k),j})}{||V_i||(M)} - \sum _k\sum _i \sum _j \alpha _i \frac{V_i(\psi _k \lambda _j(k)h(u^k_j))}{||V_i||(M)}\Big | < C/m, \end{aligned}$$

and

$$\begin{aligned} \Big | \sum _k\sum _i \sum _j \alpha _i \frac{V_i(\psi _k \lambda _j(k)h(u^k_j))}{||V_i||(M)}- \sum _i \alpha _i \frac{V_i(h)}{||V_i||(M)}\Big | < C/m, \end{aligned}$$

so we conclude

$$\begin{aligned} \Big | \sum _i \alpha _i \frac{V_i(h)}{||V_i||(M)} - \frac{\int _{M}{{\,\mathrm{Tr}\,}}h}{(n+1)\mathrm{vol}(M)}\Big | < C/m. \end{aligned}$$

In the paragraph that follows, all the integrals, traces and varifold values are computed with ${\tilde{g}}$. Each $V_i=V_{m,i}$, $i=1, \ldots , J_m=J$, is of the form

$$\begin{aligned} V_i = \sum _{q=1}^{R_{m,i}} \Sigma _{m, i,q}, \end{aligned}$$

with $R_{m,i} \in {\mathbb {N}}$, $\Sigma _{m,i,q}$ a connected, closed, smooth, embedded, minimal hypersurface of $(M,{\tilde{g}})$. Choose integers $c_{m,i}, d_m\in {\mathbb {N}}$ such that $\alpha _i=\alpha _{m,i}$ satisfies

$$\begin{aligned} |\frac{\alpha _{m,i}}{||V_{m,i}||(M)}-\frac{c_{m,i}}{d_m}|< \frac{1}{mJ_m||V_{m,i}||(M)}. \end{aligned}$$

In particular, $|1-\sum _{i=1}^{J_m}\frac{c_{m,i}||V_{m,i}||(M)}{d_m}|<1/m$ and

$$\begin{aligned} \Big | \sum _i \frac{c_{m,i}}{d_m} V_{m,i}(h) - \frac{\int _{M}{{\,\mathrm{Tr}\,}}h}{(n+1)\mathrm{vol}(M)}\Big | < C/m. \end{aligned}$$

Hence

$$\begin{aligned} \lim _{m\rightarrow \infty } \frac{\sum _{i=1}^{J_m} c_{m,i}V_{m,i}(h)}{\sum _{i=1}^{J_m}c_{m,i}||V_{m,i}||(M)} =\frac{\int _{M}{{\,\mathrm{Tr}\,}}h}{(n+1)\mathrm{vol}(M)} \end{aligned}$$

(11)

for any symmetric (0, 2)-tensor h. If we choose $h=f\cdot {\tilde{g}}$, with $f \in C^\infty (M)$, we get

$$\begin{aligned} \lim _{m\rightarrow \infty } \frac{\sum _{i=1}^{J_m}\sum _{q=1}^{R_{m,i}} c_{m,i}\int _{\Sigma _{m, i,q}} f}{\sum _{i=1}^{J_m} \sum _{q=1}^{R_{m,i}}c_{m,i}\mathrm{vol}(\Sigma _{m,i,q})} =\frac{\int _{M}f}{\mathrm{vol}(M)}. \end{aligned}$$

(12)

Because

$$\begin{aligned} V_{m,i}(h)=\sum _{q=1}^{R_{m,i}}\int _{\Sigma _{m,i,q}}h(\nu ,\nu ) = \sum _{q=1}^{R_{m,i}}\int _{\Sigma _{m,i,q}} ({{\,\mathrm{Tr}\,}}h- {{\,\mathrm{Tr}\,}}_{\Sigma _{m,i,q}}h), \end{aligned}$$

we can combine (11) with (12) to conclude

$$\begin{aligned} \lim _{m\rightarrow \infty } \frac{\sum _{i=1}^{J_m}\sum _{q=1}^{R_{m,i}} c_{m,i}\int _{\Sigma _{m, i,q}} {{\,\mathrm{Tr}\,}}_{\Sigma _{m,i,q}}h}{\sum _{i=1}^{J_m} \sum _{q=1}^{R_{m,i}}c_{m,i}\mathrm{vol}(\Sigma _{m,i,q})} =\frac{n\int _{M}{{\,\mathrm{Tr}\,}}h}{(n+1)\mathrm{vol}(M)}. \end{aligned}$$

In other words, we just proved that, assuming Property (P), one can find for a generic metric a sequence of finite lists of closed, embedded, connected minimal hypersurfaces $\big \{\Sigma _{N,1}, \ldots , \Sigma _{N, P_N}\big \}_{N\in {\mathbb {N}}}$ such that the following is true: if we denote $\int _{\Sigma _{N,i}} {{\,\mathrm{Tr}\,}}_{\Sigma _{N,i}} (h) \, d\Sigma _{N,i}$ (resp. $\mathrm{vol}(\Sigma _{N,i})$) by $X_{N,i}$ (resp. ${\bar{X}}_{N,i}$), then

$$\begin{aligned} \left| \frac{\sum _{i=1}^{P_N} X_{N,i}}{\sum _{i=1}^{P_N} {\bar{X}}_{N,i}}-\alpha \right| \le \varepsilon _N, \end{aligned}$$

(13)

where $\alpha =\frac{1}{\mathrm{vol}(M)}\int _M \frac{n{{\,\mathrm{Tr}\,}}_M h}{n+1} dM$ and $\lim _{N \rightarrow \infty }\varepsilon _N=0$. From the numbers $X_{N,i}$, ${\bar{X}}_{N,i}$, we want to construct two sequences $\{Y_j\}_{j\in {\mathbb {N}}}$ and $\{{\bar{Y}}_j\}_{j\in {\mathbb {N}}}$ such that

• for all j, there exist integers N(j), i(j) (chosen independently of h) with $Y_j = X_{N(j),i(j)}$ and ${\bar{Y}}_j = {\bar{X}}_{N(j),i(j)}$,

• moreover

  $$\begin{aligned} \lim _{q \rightarrow \infty } \frac{\sum _{j=1}^{q} Y_j}{\sum _{j=1}^{q} {\bar{Y}}_{j}}=\alpha . \end{aligned}$$

Note first that all the ${\bar{X}}_{N,i}$ are bounded below by a uniform positive constant v, according to the monotonicity formula, and that $|X_{N,i}|\le C(h) {\bar{X}}_{N,i}$ where C(h) is the maximum value that the absolute value of the trace of h can take over the Grassmannian $G_n(M)$.

Let $\{Q_N\}_{N\in {\mathbb {N}}}$ be a sequence of positive integers that will be chosen in the following order: $Q_1$ is chosen depending on $\{\Sigma _{1,i}\}$ and $\{\Sigma _{2,i}\}$, $Q_2$ is chosen depending on $Q_1$, $\{\Sigma _{1,i}\}$, $\{\Sigma _{2,i}\}$, $\{\Sigma _{3,i}\}$, and similarly $Q_{N_0}$ is chosen depending on $\{Q_1, \ldots , Q_{N_0-1}\}$, $\{\Sigma _{1,i}\}$, $\{\Sigma _{2,i}\}$, $\ldots \{\Sigma _{N_0+1,i}\}$.

If $1\le j \le Q_1P_1$, write $j=kP_1+l$ where $k\in \{0, \ldots , Q_1-1\}$ and $l\in \{1,\ldots ,P_1\}$. Then define $Y_j=X_{1,l}$ and ${\bar{Y}}_j={\bar{X}}_{1,l}$ accordingly. Notice that

$$\begin{aligned}&\left| \frac{\sum _{j=1}^{kP_1+l}Y_j}{\sum _{j=1}^{kP_1+l} {{\overline{Y}}}_j}-\alpha \right| \\&\le \left| \frac{k(\sum _{i=1}^{P_1} X_{1,i}-\alpha \sum _{i=1}^{P_1} {\overline{X}}_{1,i}) + (\sum _{i=1}^l X_{1,i}-\alpha \sum _{i=1}^l {\overline{X}}_{1,i})}{ k\sum _{i=1}^{P_1} {\overline{X}}_{1,i}+ \sum _{i=1}^l {\overline{X}}_{1,i}}\right| \\&\le \varepsilon _1+C(h)+|\alpha |, \end{aligned}$$

while

$$\begin{aligned}&\left| \frac{\sum _{j=1}^{Q_1P_1}Y_j}{\sum _{j=1}^{Q_1P_1} {{\overline{Y}}}_j}-\alpha \right| \le \varepsilon _1. \end{aligned}$$

If $Q_1P_1+1 \le j \le Q_1P_1+Q_2P_2$, we write $j=Q_1P_1+kP_2+l$ where $k\in \{0, \ldots , Q_2-1\}$ and $l\in \{1,\ldots ,P_2\}$. Then define $Y_j=X_{2,l}$ and ${\bar{Y}}_j={\bar{X}}_{2,l}$ accordingly. Now

$$\begin{aligned}&\left| \frac{\sum _{j=1}^{Q_1P_1+kP_2+l}Y_j}{\sum _{j=1}^{Q_1P_1+kP_2+l} {{\overline{Y}}}_j}-\alpha \right| \\&\le \left| \left( Q_1\left( \sum _{i=1}^{P_1} X_{1,i}-\alpha \sum _{i=1}^{P_1} {\overline{X}}_{1,i}\right) + k\left( \sum _{i=1}^{P_2} X_{2,i}-\alpha \sum _{i=1}^{P_2} {\overline{X}}_{2,i}\right) \right. \right. \\&\quad + \left. \left. \left( \sum _{i=1}^l X_{2,i}-\alpha \sum _{i=1}^l {\overline{X}}_{2,i}\right) \right) \right| \\&\quad \cdot \frac{1}{\left. (Q_1\sum _{i=1}^{P_1} {\overline{X}}_{1,i} +k\sum _{i=1}^{P_2} {\overline{X}}_{2,i}+ \sum _{i=1}^l {\overline{X}}_{2,i}\right) }\\&\le \varepsilon _1+\varepsilon _2+\frac{C(h)+|\alpha |}{Q_1P_1v} \sum _{i=1}^{P_2} {\overline{X}}_{2,i}\\&\le \varepsilon _1+\varepsilon _2+ (C(h)+|\alpha |)\varepsilon _2, \end{aligned}$$

if $Q_1$ is sufficiently large depending on $\{\Sigma _{1,i}\}$ and $\{\Sigma _{2,i}\}$, while

$$\begin{aligned}&\left| \frac{\sum _{j=1}^{Q_1P_1+Q_2P_2}Y_j}{\sum _{j=1}^{Q_1P_1+Q_2P_2}{{\overline{Y}}}_j}-\alpha \right| \le 2\varepsilon _2, \end{aligned}$$

if $Q_2$ is sufficiently large depending on $Q_1$, $\{\Sigma _{1,i}\}$ and $\{\Sigma _{2,i}\}$.

Proceeding this way we get a sequence $\{Q_N\}$ and a sequence $\{Y_j\}$ defined so that if $1+ \sum _{N=1}^{N=N_0} Q_NP_N\le j \le \sum _{N=1}^{N=N_0+1} Q_NP_N$, we write $j=\sum _{N=1}^{N=N_0} Q_NP_N+kP_{N_0+1}+l$, where $k \in \{0, \ldots , Q_{N_0+1}-1\}$ and $l \in \{1, \ldots , P_{N_0+1}\}$, and set $Y_j=\Sigma _{N_0+1,l}$, ${\bar{Y}}_j={\bar{X}}_{N_0+1,l}$. We will have

$$\begin{aligned}&\left| \frac{\sum _{j=1}^{\sum _{N=1}^{N=N_0} Q_NP_N+kP_{N_0+1}+l}Y_j}{\sum _{j=1}^{\sum _{N=1}^{N=N_0} Q_NP_N+kP_{N_0+1}+l}{{\overline{Y}}}_j}-\alpha \right| \le 2\varepsilon _{N_0}+\varepsilon _{N_0+1}+(C(h)+|\alpha |)\varepsilon _{N_0+1}, \end{aligned}$$

and

$$\begin{aligned}&\left| \frac{\sum _{j=1}^{\sum _{N=1}^{N=N_0+1} Q_NP_N}Y_j}{\sum _{j=1}^{\sum _{N=1}^{N=N_0+1} Q_NP_N}{{\overline{Y}}}_j}-\alpha \right| \le 2\varepsilon _{N_0+1}. \end{aligned}$$

This implies

$$\begin{aligned} \lim _{q \rightarrow \infty } \frac{\sum _{j=1}^{q} Y_j}{\sum _{j=1}^{q} {\overline{Y}}_{j}}=\alpha \end{aligned}$$

for any h, and we are done.

Proof of the Property (P): Let g be a smooth Riemannian metric, $\epsilon _1>0$ and ${\tilde{K}}>0$ be constants, and choose

$$\begin{aligned} S=(K, \{{\hat{B}}_k\}, \{B_k\}, \{\phi _k\}, \{q_k\}, \{x^k_l\}) \in {\mathcal {C}}_{g, {\tilde{K}}, \epsilon _1}. \end{aligned}$$

Let $\mathcal {U}$ be a $C^\infty$ neighborhood of g. Let $N=KL(n+1)$. Choose $\epsilon '>0$ sufficiently small and $q\ge N+3$ sufficiently large so that if $g'\in \Gamma _\infty$ satisfies $||g-g'||_{C^q}<\epsilon '$, then $g'\in \mathcal {U}$. For each k, l, j we associate a variable $t_{k,l,j}\in [0,1]$ and we order them by lexicographical order on the indices. We can find a smooth (0, 2)-tensor ${\bar{h}}^k_{l,j}$ so that $||{\bar{h}}^k_{l,j}-h^k_{l,j}||_{C^q}<\epsilon '$ and such that $\{{\bar{h}}^k_{l,j}\}_{l,j}$ is linearly independent in a neighborhood of $q_k$ where $\phi _k$ is equal to 1 and $\phi _{k'}$ is zero for $k'\ne k$.

Consider the following N-parameter family of metrics. For a $t=(t_{k,l,j})\in [0,1]^N$, we define

$$\begin{aligned} {\hat{g}}(t)= g+2\sum _{k,l,j} \psi _k t_{k,l,j} {\bar{h}}^k_{l,j}. \end{aligned}$$

As t goes to zero, we have the following expansion

$$\begin{aligned}&\mathrm{vol}(M,{\hat{g}}(t))^{\frac{n}{n+1}} = \mathrm{vol}_g(M)^{\frac{n}{n+1}} \nonumber \\&+ \frac{n}{(n+1)} {\mathrm{vol}_g(M)^{-\frac{1}{n+1}}}\sum _{k,l,j} t_{k,l,j}\int _M \psi _k dv_g + o(||t||_1) + O(\epsilon '||t||_1), \qquad \end{aligned}$$

(14)

where $||t||_1=\sum _{k,l,j} |t_{k,l,j}|$. Also

$$\begin{aligned} \frac{\partial }{\partial t_{k,l,j}} \mathrm{vol}(M,{\hat{g}}(t)) = \int _M \psi _k {{\,\mathrm{Tr}\,}}_{{\hat{g}}(t)}({\bar{h}}^k_{l,j}) dv_{{\hat{g}}(t)} =\int _M \psi _k dv_g + o(1)+O(\epsilon '). \end{aligned}$$

We will say that a function $f:[0,\delta ]^N\rightarrow {\mathbb {R}}$ is $\epsilon '$-close to another function $g:[0,\delta ]^N\rightarrow {\mathbb {R}}$ if, when appropriately rescaled to be functions defined on $[0,1]^N$, they are at distance less than $\epsilon '$ in the $L^\infty$ norm, i.e.

$$\begin{aligned} ||\frac{1}{\delta }f_\delta -\frac{1}{\delta }g_\delta ||_\infty <\epsilon ' \end{aligned}$$

with $f_\delta (s)= f(\delta s)$ and $g_\delta (s)=g(\delta s)$. By (14), the function

$$\begin{aligned} f_0(t) : = \frac{\mathrm{vol}(M,{\hat{g}}(t))^{n/(n+1)}}{\mathrm{vol}_g(M)^{n/(n+1)}} - \frac{n}{(n+1)}\frac{1}{\mathrm{vol}_g(M)}\sum _{k,l,j} t_{k,l,j}\int _M \psi _k dv_g \end{aligned}$$

is $C\epsilon '$-close to the constant function equal to 1 on $[0,\delta ]^N$, where $C=C(g)$ depends only on g and might differ from line to line, if $\delta$ is sufficiently small.

If $\delta >0$ is sufficiently small, we also have that ${\hat{g}}:[0,\delta ]^N \rightarrow \Gamma _q$ is an embedding and $||{\hat{g}}(t)-g||_{C^q}<\epsilon '/2$ for every $t\in [0,\delta ]^N$. We can slightly perturb ${\hat{g}}$ in the $C^\infty$ topology into a $C^\infty$ map $g':[0,\delta ]^N \rightarrow \Gamma _q$ so that the conclusion of Lemma 2 is satisfied. In particular, we can assume $||g'(t)-{\hat{g}}(t)||_{C^q}<\epsilon '/4$ and $||\frac{\partial g'}{\partial v}(t)-\frac{\partial {\hat{g}}}{\partial v}(t)||_{C^q} < \epsilon '/4$ for any $t \in [0,\delta ]^N$ and $v\in {\mathbb {R}}^N$, $|v|=1$, and the function

$$\begin{aligned} f_1(t) : = \frac{\mathrm{vol}(M,g'(t))^{n/(n+1)}}{\mathrm{vol}_g(M)^{n/(n+1)}} - \frac{n}{(n+1)}\frac{1}{\mathrm{vol}_g(M)}\sum _{k,l,j} t_{k,l,j}\int _M \psi _k dv_g \end{aligned}$$

is $C\epsilon '$-close to the constant function equal to 1 on $[0,\delta ]^N$.

The normalized widths $p^{-\frac{1}{n+1}}\omega _p(g'(t))$ of $g'(t)$ ($t\in [0,\delta ]^N$) are uniformly Lipschitz continuous on $[0,\delta ]^N$ by Lemma 1. Hence, by the Weyl Law for the Volume Spectrum [9], the functions $t \mapsto p^{-\frac{1}{n+1}}\omega _p(g'(t))$ converge uniformly to the function $t \mapsto a(n){{\,\mathrm{Vol}\,}}(M,g'(t))^{\frac{n}{n+1}}$. Hence if p is sufficiently large, $|p^{-\frac{1}{n+1}}\omega _p(g'(t)) - a(n){{\,\mathrm{Vol}\,}}(M,g'(t))^{\frac{n}{n+1}}|<\delta \epsilon '$ and the function

$$\begin{aligned} f_2(t) : = \frac{\omega _p(g'(t))}{a(n)\mathrm{vol}_g(M)^{n/(n+1)}p^{1/(n+1)}} - \frac{n}{(n+1)}\frac{1}{\mathrm{vol}_g(M)}\sum _{k,l,j} t_{k,l,j}\int _M \psi _k dv_g \end{aligned}$$

is $C\epsilon '$-close to the constant function equal to 1 on $[0,\delta ]^N$.

Then at each $t\in {\mathcal {A}}$ (where ${\mathcal {A}}$ is given by Lemma 2), there is a varifold $V\in {\mathcal {V}}(g'(t))$ with support a minimal hypersurface $\Sigma$ such that

$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial t_{k,l,j}} p^{-\frac{1}{n+1}}\omega _p({g'}(t))&= p^{-\frac{1}{n+1}} \frac{\partial }{\partial t_{k,l,j}} ||V||(M,g'(t)) \\&=p^{-\frac{1}{n+1}}||V||(\psi _k {{\,\mathrm{Tr}\,}}_\Sigma ({{\bar{h}}^k_{l,j}}))+O(\epsilon ') \\&=p^{-\frac{1}{n+1}}\big (||V||(\psi _k{{\,\mathrm{Tr}\,}}_{M,g'(t)}{\bar{h}}^k_{l,j} ) -V(\psi _k{{\bar{h}}^k_{l,j}}) \big )+ O(\epsilon ')\\&=p^{-\frac{1}{n+1}}\big (||V||(\psi _k) -V(\psi _k{h^k_{l,j}}) \big )+O(\epsilon '), \end{aligned} \end{aligned}$$

(15)

where ||V||(.), V(.) and the traces are computed with respect to $g'(t)$.

Given $\eta >0$, we can choose $0<\epsilon '<\eta$ sufficiently small compared to $C=C(g)$ so that we can apply Lemma 3 to $f_2$ and to ${\mathcal {A}}$. We get sequences of points $\{y_{1,m}\}_m, \ldots , \{y_{N+1,m}\}$ in ${\mathcal {A}}$ converging to a common limit $y \in (0,\delta )^N$ such that the gradients $\nabla f_2(y_{i,m})$ converge to $N+1$ vectors $v_1,\ldots ,v_{N+1}$ with

$$\begin{aligned} d_{{\mathbb {R}}^N}\left( 0, {{\,\mathrm{Conv}\,}}(v_1,\ldots ,v_{N+1})\right) < \eta . \end{aligned}$$

Let $\{\alpha _1, \ldots , \alpha _{N+1}\} \subset [0,1]$ with $\sum _{i=1}^{N+1} \alpha _i=1$ such that $|\alpha _1v_1 + \cdots + \alpha _{N+1}v_{N+1}|< \eta$. Then for sufficiently large m, we have

$$\begin{aligned} |\alpha _1\nabla f_2(y_{1,m}) + \cdots + \alpha _{N+1} \nabla f_2(y_{N+1,m})|< \eta , \end{aligned}$$

and hence

$$\begin{aligned} |\alpha _1\frac{\partial f_2}{\partial t_{k,l,j}}(y_{1,m}) + \cdots + \alpha _{N+1}\frac{\partial f_2}{\partial t_{k,l,j}}(y_{N+1,m})|< \eta \end{aligned}$$

(16)

for all k, l, j.

According to Lemma 2, each gradient based at $y_{i,m}$ corresponds to a varifold of mass $\omega _p(g'(y_{i,m}))$ whose support is a minimal hypersurface in $(M,g'(y_{i,m}))$ of index bounded by p. Hence for all i, by Sharp’s Compactness Theorem [20] a subsequence in m of these varifolds converges to a varifold of ${\mathcal {V}}(g'(y))$ whose mass is $\omega _p(g'(y))$. By (15) and (16), we have $N+1$ varifolds $V_i$ in ${\mathcal {V}}(g'(y))$ such that

$$\begin{aligned} \quad \left| \sum _i \alpha _i \frac{||V_i||(\psi _k)-V_i(\psi _kh^k_{l,j})}{a(n)\mathrm{vol}_g(M)^{n/(n+1)}p^{1/(n+1)}} - \frac{n}{(n+1)}\frac{1}{\mathrm{vol}_g(M)}\int _M \psi _k dv_g\right| < C\eta \end{aligned}$$

for all k, l, j, where $||V_i||(.)$ and $V_i(.)$ are computed with respect to $g'(y)$. This implies

$$\begin{aligned} \forall k,l,j \quad \left| \sum _i \alpha _i \frac{||V_i||(\psi _k)-V_i(\psi _kh^k_{l,j})}{||V_i||(M)} - \frac{n}{(n+1)}\frac{1}{\mathrm{vol}_g(M)}\int _M \psi _k dv_g\right| < C\eta . \end{aligned}$$

We also have that for all i, k, l, one has

$$\begin{aligned}&\left| \sum _{j=1}^{n+1} V_i(\psi _kh^k_{l,j})-||V_i||(\psi _k)\right| = \left| V_i(\psi _k g) - ||V_i||(\psi _k)\right|< C\eta ||V_i||(M),\\&\quad \text { and } \left| \frac{1}{(n+1)}\frac{1}{\mathrm{vol}_g(M)}\int _M \psi _k dv_g - \frac{1}{(n+1)}\frac{1}{\mathrm{vol}_{g'(y)}(M)}\int _M \psi _k dv_{g'(y)} \right| <C\eta . \end{aligned}$$

We deduce the following:

$$\begin{aligned} \forall k,l,j \quad \left| \sum _i \alpha _i \frac{V_i(\psi _kh^k_{l,j})}{||V_i||(M)} - \frac{1}{(n+1)}\frac{1}{\mathrm{vol}_{g'(y)}(M)}\int _M \psi _k dv_{g'(y)}\right| < C\eta . \end{aligned}$$

(17)

The metric $g'(y)\in \Gamma _q$ satisfies $||g'(y)-g||_{C^q}<3\epsilon '/4$. We apply Lemma 4 to $\bigcup _{i=1}^{N+1}{{\,\mathrm{spt}\,}}(V_i)$ and find a $C^q$ metric $\overline{g}$ such that $||\overline{g}-g||_{C^q}<4\epsilon '/5$, each ${{\,\mathrm{spt}\,}}(V_i)$ is nondegenerate minimal with respect to $\overline{g}$ and (17) is still valid with $g'(y)$ replaced by $\overline{g}$. If ${\tilde{g}}$ is a $C^\infty$ metric that is sufficiently close to $\overline{g}$ in the $C^q$ topology, then ${\tilde{g}} \in \mathcal {U}$. Because of the nondegeneracy of ${{\,\mathrm{spt}\,}}{V_i}$ with respect to $\overline{g}$ and the Implicit Function Theorem, we can also assure that there are varifolds $V_1,\ldots ,V_J$ of ${\mathcal {V}}({\tilde{g}})$ whose support ${{\,\mathrm{spt}\,}}(V_j)$ are nondegenerate, and coefficients $\alpha _1,\ldots ,\alpha _J\in [0,1]$ with $\sum _i \alpha _i =1$ satisfying

$$\begin{aligned} \forall k,l,j \quad \left| \sum _i \alpha _i \frac{V_i(\psi _k h^k_{l,j})}{||V_i||(M)} - \frac{1}{(n+1)}\frac{1}{\mathrm{vol}_{{\tilde{g}}}(M)}\int _M \psi _k dv_{{\tilde{g}}}\right| < C\eta , \end{aligned}$$

(18)

where the terms of the sum are computed for the metric ${\tilde{g}}$. Since $\eta$ is arbitrarily small, we have proved Property (P).