Abstract
We explore to what extent one may hope to preserve geometric properties of three-dimensional manifolds with lower scalar curvature bounds under Gromov–Hausdorff and Intrinsic Flat limits. We introduce a new construction, called sewing, of three-dimensional manifolds that preserves positive scalar curvature. We then use sewing to produce sequences of such manifolds which converge to spaces that fail to have nonnegative scalar curvature in a standard generalized sense. Since the notion of nonnegative scalar curvature is not strong enough to persist alone, we propose that one pair a lower scalar curvature bound with a lower bound on the area of a closed minimal surface when taking sequences as this will exclude the possibility of sewing of manifolds.
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1 Introduction
In this paper, we study three-dimensional manifolds with positive scalar curvature. The scalar curvature of a Riemannian manifold is the average of the Ricci curvatures which in turn is the average of the sectional curvatures. It can be determined more simply by taking the following limit:
where \({\text {Vol}}_{{\mathbb {E}}^3}(B(0,r))=(4/3) \pi r^3\) and \({\text {Vol}}_{M^3}(B(p, r))\) is the Hausdorff measure of the ball about p of radius r in our manifold, \(M^3\).
In [16], Gromov asks the following pair of deliberately vague questions which we paraphrase here: Given a class of Riemannian manifolds, \({\mathcal {B}}\), what is the weakest notion of convergence such that a sequence of manifolds, \(M_j \in {\mathcal {B}}\), subconverges to a limit\(M_\infty \in {\mathcal {B}}\)where now we will expand\({\mathcal {B}}\)to include singular metric spaces? What is this generalized class of singular metrics spaces that should be included in\({\mathcal {B}}\)? Gromov points out that when \({\mathcal {B}}\) is the class of Riemannian manifolds with nonnegative sectional curvature then the “best known” answer to this question is Gromov–Hausdorff convergence and the singular limit spaces are then Alexandrov spaces with nonnegative Alexandrov curvature. When \({\mathcal {B}}\) is the class of Riemannian manifolds with nonnegative Ricci curvature, one uses Gromov–Hausdorff and metric measure convergence to obtain limits which are metric measure spaces with generalized nonnegative Ricci curvature as in work of Cheeger–Colding [8]. Work towards defining classes of singular metric measure spaces with generalized notions of nonnegative Ricci has been completed by Ambrosio-Gigli-Savare, Lott-Villani, Sturm and others [1, 21, 28].
Gromov then writes that “the most tantalizing relation\({\mathcal {B}}\)is expressed with the scalar curvature by\(\mathrm{Scal}\ge k\)” [16]. Bamler [4] and Gromov [15] have proven that under \(C^0\) convergence to smooth Riemannian limits \(\mathrm{Scal}\ge 0\) is preserved. In order to find the weakest notion of convergence which preserves \(\mathrm{Scal}\ge 0\) in some sense, Gromov has suggested that one might investigate intrinsic flat convergence [16]. The intrinsic flat distance was first defined in work of the third author with Wenger [31], who also proved that for noncollapsing sequences of manifolds with nonnegative Ricci curvature, intrinsic flat limits agree with Gromov–Hausdorff and metric measure limits [30]. Intrinsic flat convergence is a weaker notion of convergence in the sense that there are sequences of manifolds with no Gromov–Hausdorff limit that have intrinsic flat limits, including Ilmanen’s example of a sequence of three spheres with positive scalar curvature [31]. The third author has investigated intrinsic flat limits of manifolds with nonnegative scalar curvature under additional conditions with Lee, Huang, LeFloch and Stavrov [17, 19, 20, 27]. These papers support Gromov’s suggestion in the sense that the limits obtained in these papers have generalized nonnegative scalar curvature.
Here we construct a sequence of Riemannian manifolds, \(M_j^{3}\), with positive scalar curvature that converges in the intrinsic flat, metric measure and Gromov–Hausdorff sense to a singular limit space, Y, which fails to satisfy (1) [Example 6.1]. In fact, the limit space is a sphere with a pulled thread:
where C is one geodesic in \({\mathbb {S}}^3\) (see Sect. 4). The scalar curvature about the point \(p_0 =[C(t)]\) formed from the pulled thread is computed in Lemma 6.3 to be
In this sense, the limit space does not have generalized nonnegative scalar curvature.
We construct our sequence using a new method we call sewing developed in Propositions 3.1–3.3. Before we can sew the manifolds, the first two authors construct short tunnels between points in the manifolds building on prior work of Gromov–Lawson [12] and Schoen–Yau [32]. The details of this construction are in the Appendix. In a subsequent paper [7], we will extend this sewing technique to also provide examples whose limit spaces fail to satisfy the Scalar Torus Rigidity Theorem [12, 32] and the Positive Mass Rigidity Theorem [33]. These examples, all constructed using the sewing techniques developed in this paper, demonstrate that Gromov–Hausdorff and Intrinsic Flat limit spaces of noncollapsing sequences of manifolds with positive scalar curvature may fail to satisfy key properties of nonnegative scalar curvature.
In light of these counter examples and the aforementioned positive results towards Gromov’s conjecture, the third author has suggested in [26] to adapt the class \({\mathcal {B}}\). There it is proposed that the initial class of smooth Riemannian manifolds in \({\mathcal {B}}\) should have nonnegative scalar curvature, a uniform lower bound on volume (as assumed implicitly by Gromov), and also a uniform lower bound on the minimal area of a closed minimal surface in the manifold, \(\text {MinA}(M)\). The sequences of \(M_j^{3}\) we construct using our new sewing methods have positive scalar curvature and a uniform lower bound on volume, but \(\text {MinA}(M_j)\rightarrow 0\). Intuitive reasons as to why a uniform lower bound on \(\text {MinA}(M_j)\) is a natural condition are described in [26] along with a collection of related conjectures and open problems. Here we will simply propose the following possible revision of Gromov’s vague conjecture:
Conjecture 1.1
Suppose a sequence of Riemannian manifolds, \(M^3_j\), have
and the sequence converges in the intrinsic flat sense, \(M_j {\mathop {\longrightarrow }\limits ^{{\mathcal {F}}}} M_\infty \).
Then at every point \(p\in M_\infty \) we have
This paper is part of the work towards Jorge Basilio’s doctoral dissertation at the CUNY Graduate Center conducted under the advisement of Professors Józef Dodziuk and Christina Sormani. We would like to thank Jeff Jauregui, Marcus Khuri, Sajjad Lakzian, Dan Lee, Raquel Perales, Conrad Plaut, and Catherine Searle for their interest in this work.
2 Background
In this section, we first briefly review Gromov–Lawson and Schoen–Yau’s work. We then review Gromov–Hausdorff, Metric Measure, and Intrinsic Flat Convergence covering the key definitions as well as theorems applied in this paper to prove our example converges with respect to all three notions of convergence.
2.1 Gluing Gromov–Lawson and Schoen–Yau Tunnels
Using different techniques, Gromov–Lawson and Schoen–Yau described how to construct tunnels diffeomorphic to \({{\mathbb {S}}}^2 \times [0,1]\) with metric tensors of positive scalar curvature that can be glued smoothly into three-dimensional spheres of constant sectional curvature [12, 32]. See Fig. 1. These tunnels are the first crucial piece for our construction.
The tunnel
Here we need to explicitly estimate the volume and diameter of these tunnels. So the first and second authors prove the following lemma in the appendix.
Lemma 2.1
Let \(0<\delta /2 < 1\). Given a complete Riemannian manifold, \(M^3\), that contains two balls \(B(p_i,\delta /2)\subset M^3\), \(i=1,2\), with constant positive sectional curvature \(K \in (0,1]\) on the balls, and given any \(\epsilon >0\), there exists a \(\delta _0>0\) sufficiently small so that we may create a new complete Riemannian manifold, \(N^3\), in which we remove two balls and glue in a cylindrical region, U, between them:
where \(U=U(\delta _0)\) has a metric of positive scalar curvature (See Fig. 1) with
where
hence,
The collars \(C_i= B(p_i,\delta /2) \setminus B(p_i,\delta _0)\) identified with subsets of \(N^3\) have the original metric of constant curvature and the tunnel \(U'=U\setminus (C_1\cup C_2)\) has arbitrarily small diameter \(O(\delta _0)\) and volume \(O(\delta _0^3)\). Therefore with appropriate choice of \(\delta _0\), we have
and
We note that if \(M^3\) has positive scalar curvature then so does \(N^3\) and that, after inserting the tunnel, \(\partial B(p_1,\delta /2)\) and \(\partial B(p_2,\delta /2)\) are arbitrarily close together because of (9). Note that we have restricted to three dimensions here and required constant sectional curvature on the balls for simplicity. The first two authors will generalize these conditions in future work. This lemma suffices for proving all the examples in this paper.
2.2 Review GH Convergence
Gromov introduced the Gromov–Hausdorff distance in [14].
First recall that \(\varphi : X \rightarrow Y\) is distance preserving iff
This is referred to as a metric isometric embedding in [19] and is distinct from a Riemannian isometric embedding.
Definition 2.2
(Gromov) The Gromov–Hausdorff distance between two compact metric spaces \(\left( X, d_X\right) \) and \(\left( Y, d_Y\right) \) is defined as
where Z is a complete metric space, and \(\varphi : X \rightarrow Z\) and \(\psi :Y\rightarrow Z\) are distance preserving maps and where the Hausdorff distance in Z is defined as
Gromov proved that this is indeed a distance on compact metric spaces: \(d_{GH}\left( X,Y\right) =0\) iff there is an isometry between X and Y. When studying metric spaces which are only precompact, one may take their metric completions before studying the Gromov–Hausdorff distance between them.
We write
Gromov proved that if \(X_j {\mathop {\longrightarrow }\limits ^{ \text {GH}}} X_\infty \) then there is a common compact metric space Z and distance preserving maps \(\varphi _j: X_j \rightarrow Z\) such that
We say \(p_j \in X_j\) converges to \(p_\infty \in X_\infty \) if there is such a set of maps such that \(\varphi _j(p_j)\) converges to \(\varphi _\infty (p_\infty )\) as points in Z. These limits are not uniquely defined but they are useful and every point in the limit space is a limit of such a sequence in this sense.
Theorem 2.3
(Gromov) Suppose \(\epsilon _j \rightarrow 0\). If a sequence of metric spaces \((X_j, d_j)\) have \(\epsilon _j\) almost isometries
such that
and
then
Note that \(p_j \in X_j\) converges to \(p_\infty \in X_\infty \) if \(F_j(p_j) \rightarrow p_\infty \in X_\infty \).
Gromov’s Compactness Theorem states that a sequence of manifolds with nonnegative Ricci (or Sectional) Curvature, and a uniform upper bound on diameter, has a subsequence which converges in the Gromov–Hausdorff sense to a geodesic metric space [14]. If a sequence of manifolds has nonnegative sectional curvature, then they satisfy the Toponogov Triangle Comparison Theorem. Taking the limits of the points in the triangles, one sees that the Gromov–Hausdorff limit of the sequence also satisfies the triangle comparison. Thus the limit spaces are Alexandrov spaces with nonnegative Alexandrov curvature (cf. [5]).
2.3 Review of Metric Measure Convergence
Fukaya introduced the notion of metric measure convergence of metric measure spaces \((X_j, d_j, \mu _j)\) in [10]. He assumed the sequence converged in the Gromov–Hausdorff sense as in (16) and then required that the push forwards of the measures converge as well,
Cheeger–Colding proved metric measure convergence of noncollapsing sequences of manifolds with Ricci uniformly bounded below in [8] where the measure on the limit is the Hausdorff measure. They proved metric measure convergence by constructing almost isometries and showing the Hausdorff measures of balls about converging points converge:
They also studied collapsing sequences obtaining metric measure convergence to other measures on the limit space. Cheeger and Colding applied this metric measure convergence to prove that limits of manifolds with nonnegative Ricci curvature have generalized nonnegative Ricci curvature. In particular they prove the limits satisfy the Bishop–Gromov Volume Comparison Theorem and the Cheeger–Gromoll Splitting Theorem.
Sturm, Lott, and Villani then developed the CD(k,n) notion of generalized Ricci curvature on metric measure spaces in [21, 28]. In [29], Sturm extended the study of metric measure convergence beyond the consideration of sequences of manifolds which already converge in the Gromov–Hausdorff sense, using the Wasserstein distance. This is also explored in Villani’s text [34]. CD(k,n) spaces converge in this sense to CD(k,n) spaces. RCD(k,n) spaces developed by Ambrosio-Gigli-Savare are also preserved under this convergence [1]. RCD(k,n) spaces are CD(k,n) spaces which also require that the tangent cones almost everywhere are Hilbertian. There has been significant work studying both of these classes of spaces proving they satisfy many of the properties of Riemannian manifolds with lower bounds on their Ricci curvature.
2.4 Review of Integral Current Spaces
The Intrinsic Flat Distance is defined and studied in [31] by applying sophisticated ideas of Ambrosio–Kirchheim [2] extending earlier work of Federer–Fleming [9]. Limits of Riemannian manifolds under intrinsic flat convergence are integral current spaces, a notion introduced by the third author and Stefan Wenger in [31].
Recall that Federer–Fleming first defined the notion of an integral current as an extension of the notion of a submanifold of Euclidean space [9]. That is a submanifold \(\psi : M^m \rightarrow {\mathbb {E}}^N\) can be viewed as a current \(T=\psi _{\#}[M ]\) acting on m-forms as follows:
If \(\omega = f\,d\pi _1 \wedge \cdots \wedge d\pi _m\) then
They define boundaries of currents as \(\partial T(\omega ) = T (d\omega )\) so that then the boundary of a submanifold with boundary is exactly what it should be. They define integer rectifiable currents more generally as countable sums of images under Lipschitz maps of Borel sets. The integral currents are integer rectifiable currents whose boundaries are integer rectifiable.
Ambrosio–Kirchheim extended the notion of integral currents to arbitrary complete metric space [2]. As there are no forms on metric spaces, they use deGeorgi’s tuples of Lipschitz functions,
This integral is well defined because Lipschitz functions are differentiable almost everywhere. They define boundary as follows:
which matches with
They also define integer rectifiable currents more generally as countable sums of images under Lipschitz maps of Borel sets. The integral currents are integer rectifiable currents whose boundaries are integer rectifiable.
The notion of an integral current space was introduced in [31].
Definition 2.4
An m-dimensional integral current space, \(\left( X,d, T\right) \), is a metric space, (X, d) with an integral current structure \(T \in {{\mathbf {I}}}_m\left( {\bar{X}}\right) \) where \({\bar{X}}\) is the metric completion of X and \(\mathrm{set}(\hbox {T})=\hbox {X}\). Given an integral current space \(M=\left( X,d,T\right) \) we will use \(\mathrm{set}\left( \hbox {M}\right) \) or \(X_M\) to denote X, \(d_M=d\) and \([\![M ]\!]=T \). Note that \(\mathrm{set}\left( \partial \hbox {T}\right) \subset {\bar{\hbox {X}}}\). The boundary of \(\left( X,d,T\right) \) is then the integral current space:
If \(\partial T=0\) then we say \(\left( X,d,T\right) \) is an integral current without boundary.
A compact-oriented Riemannian manifold with boundary, \(M^m\), is an integral current space, where \(X=M^m\), d is the standard metric on M and T is integration over M. In this case \({{\mathbf {M}}}(M)={\text {Vol}}(M)\) and \(\partial M\) is the boundary manifold. When M has no boundary, \(\partial M=0\).
Ambrosio–Kirchheim defined the mass \({{\mathbf {M}}}(T)\) and the mass measure ||T|| of a current in [2]. We apply the same notions to define a mass for an integral current space. Applying their theorems we have
where \(\lambda (x)\) is the area factor and \(\theta _T\) is the weight. In particular, \(\lambda (x)=1\) when the tangent cone at x is Euclidean which is true on a Riemannian manifold where the weight is also 1. This is true almost everywhere in the examples in this paper as well. The mass measure, ||T||, is a measure on X and satisfies
2.5 Review of the Intrinsic Flat Distance
The Intrinsic Flat distance was defined in work of the third author and Stefan Wenger [31] as a new distance between Riemannian manifolds based upon the Federer–Fleming flat distance [9] and the Gromov–Hausdorff distance [14].
Recall that the Federer–Fleming flat distance between m-dimensional integral currents \(S,T\in {{\mathbf {I}}}_m\left( Z\right) \) is given by
where \(U\in {{\mathbf {I}}}_m\left( Z\right) \) and \(V\in {{\mathbf {I}}}_{m+1}\left( Z\right) \).
In [31], the third author and Wenger imitate Gromov’s definition of the Gromov–Hausdorff distance (which he called the intrinsic Hausdorff distance) by replaced the Hausdorff distance by the Flat distance:
Definition 2.5
(Sormani and Wenger [31]) For \(M_1=\left( X_1,d_1,T_1\right) \) and \(M_2=\left( X_2,d_2,T_2\right) \in {\mathcal {M}}^m\) let the intrinsic flat distance be defined:
where the infimum is taken over all complete metric spaces \(\left( Z,d\right) \) and distance preserving maps \(\varphi _1 : \left( {\bar{X}}_1,d_1\right) \rightarrow \left( Z,d\right) \) and \(\varphi _2: \left( {\bar{X}}_2,d_2\right) \rightarrow \left( Z,d\right) \) and the flat norm \(d_F^Z\) is taken in Z. Here \({\bar{X}}_i\) denotes the metric completion of \(X_i\) and \(d_i\) is the extension of \(d_i\) on \({\bar{X}}_i\), while \(\varphi _\# T\) denotes the push forward of T.
They then prove that this distance is 0 iff the spaces are isometric with a current preserving isometry. They say
And prove that this happens iff there is a complete metric space Z and distance preserving maps \(\varphi _j: M_j \rightarrow Z\) such that
Note that in contrast to Gromov’s embedding theorem as stated in (16), the Z here is only complete and not compact.
There is a special integral current space called the zero space,
Following the definition above, \(M_j {\mathop {\longrightarrow }\limits ^{{\mathcal {F}}}} \mathbf{{0}}\) iff \(d_{{\mathcal {F}}}(M_j, \mathbf{{0}}) \rightarrow 0\) which implies there is a complete metric space Z and distance preserving maps \(\varphi _j: M_j \rightarrow Z\) such that
Note that in this case the manifolds disappear and points have no limits.
Combining Gromov’s Embedding Theorem with Ambrosio–Kirchheim’s Compactness Theorem one has:
Theorem 2.6
(Sormani and Wenger [31]) Given a sequence of m-dimensional integral current spaces \(M_j=\left( X_j, d_j, T_j\right) \) such that \(X_j\) are equibounded and equicompact and with uniform upper bounds on mass and boundary mass. A subsequence converges in the Gromov–Hausdorff sense \(\left( X_{j_i}, d_{j_i}\right) {\mathop {\longrightarrow }\limits ^{ \text {GH}}} \left( Y,d_Y\right) \) and in the intrinsic flat sense \(\left( X_{j_i}, d_{j_i}, T_{j_i}\right) {\mathop {\longrightarrow }\limits ^{{\mathcal {F}}}} \left( X,d,T\right) \), where either \(\left( X,d,T\right) \) is an m-dimensional integral current space with \(X \subset Y\) or it is the \(\mathbf{0}\) current space.
Note that in [30], the third author and Wenger prove if the \(M_j\) have nonnegative Ricci curvature then in fact the intrinsic flat and Gromov–Hausdorff limits agree. Matveev and Portegies have extended this to more general lower bounds on Ricci curvature in [22]. With only lower bounds on scalar curvature the limits need not agree as seen in the Appendix of [31]. There are also sequences of manifolds with nonnegative scalar curvature that have no Gromov–Hausdorff limit but do converge in the intrinsic flat sense (cf. Ilmanen’s example presented in [31] and also [18]).
In [35], Wenger proved that any sequence of Riemannian manifolds with a uniform upper bound on diameter, volume, and boundary volume has a subsequence which converges in the intrinsic flat sense to an integral current space (cf. [31]). It is possible that the limit space is just the \(\mathbf {0}\) space which happens for example when the volumes of the manifolds converge to 0.
Note that when \(M_j {\mathop {\longrightarrow }\limits ^{{\mathcal {F}}}} M_\infty \) the masses are lower semicontinuous:
where the mass of an integral current space is just the mass of the integral current structure. The mass is just the volume when M is a Riemannian manifold and can be computed using (29) otherwise. As there is not equality here, intrinsic flat convergence does not imply metric measure convergence.
In [23], Portegies has proven that when a sequence converges in the intrinsic flat sense and in addition \({{\mathbf {M}}}(M_j)\) is assumed to converge to \({{\mathbf {M}}}(M_\infty )\), then the spaces do converge in the metric measure sense, where the measures are taken to be the mass measures.
2.6 Useful Lemmas and Theorems Concerning Intrinsic Flat Convergence
The following lemmas, definitions, and theorems appear in work of the third author [25], although a few (labeled only as c.f. [25]) were used within proofs in older work of the third author with Wenger [30]. All are proven rigorously in [25].
Lemma 2.7
(c.f. Sormani [25]) A ball in an integral current space, \(M=\left( X,d,T\right) \), with the current restricted from the current structure of the Riemannian manifold is an integral current space itself,
for almost every \(r > 0\). Furthermore,
Lemma 2.8
(c.f. Sormani [25]) When M is a Riemannian manifold with boundary
is an integral current space for all \(r > 0\).
Definition 2.9
(c.f. Sormani [25]) If \(M_i=(X_i, d_i,T_i) {\mathop {\longrightarrow }\limits ^{{\mathcal {F}}}} M_\infty =(X_\infty , d_\infty ,T_\infty )\), then we say \(x_i\in X_i\) are a converging sequence that converge to \(x_\infty \in {\bar{X}}_\infty \) if there exists a complete metric space Z and distance preserving maps \(\varphi _i:X_i\rightarrow Z\) such that
If we say collection of points, \(\{p_{1,i}, p_{2,i},\ldots ,p_{k,i}\}\), converges to a corresponding collection of points, \(\{p_{1,\infty }, p_{2,\infty },\ldots ,p_{k,\infty }\}\), if \(\varphi _{i}(p_{j,i}) \rightarrow \varphi _\infty (p_{j, \infty })\) for \(j=1,\ldots ,k\).
Definition 2.10
(c.f. Sormani [25] ) If \(M_i=(X_i, d_i,T_i) {\mathop {\longrightarrow }\limits ^{{\mathcal {F}}}} M_\infty =(X_\infty , d_\infty ,T_\infty )\), then we say \(x_i\in X_i\) are Cauchy if there exists a complete metric space Z and distance preserving maps \(\varphi _i:M_i\rightarrow Z\) such that
We say the sequence is disappearing if \(z_\infty \notin \varphi _\infty (X_\infty )\). We say the sequence has no limit in \({\bar{X}}_\infty \) if \(z_\infty \notin \varphi _\infty ({\bar{X}}_\infty )\).
Lemma 2.11
(c.f. Sormani [25]) If a sequence of integral current spaces, \(M_i=\left( X_i,d_i,T_i\right) \in {\mathcal {M}}_0^m\), converges to an integral current space, \(M=\left( X,d,T\right) \in {\mathcal {M}}_0^m\), in the intrinsic flat sense, then every point x in the limit space X is the limit of points \(x_i\in M_i\). In fact, there exists a sequence of maps \(F_i: X \rightarrow X_i\) such that \(x_i=F_i(x)\) converges to x and
Lemma 2.12
(c.f. Sormani [25]) If \(M_j {\mathop {\longrightarrow }\limits ^{{\mathcal {F}}}} M_\infty \) and \(p_j \rightarrow p_\infty \in {\bar{X}}_\infty \), then for almost every \(r_\infty >0\) there exists a subsequence of \(M_j\) also denoted \(M_j\) such that
are integral current spaces for \(j\in \{1,2,\ldots ,\infty \}\) and we have
If \(p_j\) are Cauchy with no limit in \({\bar{X}}_\infty \) then there exists \(\delta >0\) such that for almost every \(r \in (0,\delta )\) such that \(S(p_j,r)\) are integral current spaces for \(j\in \{1,2,\ldots \}\) and we have
If \(M_j {\mathop {\longrightarrow }\limits ^{{\mathcal {F}}}} \mathbf {0}\) then for almost every r and for all sequences \(p_j\) we have (46).
Theorem 2.13
(c.f. Sormani [25]) Suppose \(M_i=(X_i, d_i, T_i)\) are integral current spaces and
and suppose we have Lipschitz maps into a compact metric space Z,
then a subsequence converges to a Lipschitz map
More specifically, there exists distance preserving maps of the subsequence, \(\varphi _i: X_i \rightarrow Z\), such that
and for any sequence \(p_i\in X_i\) converging to \(p\in X_\infty \) (i.e., \(d_Z(\varphi _i(p_i), \varphi _\infty (p))\rightarrow 0\)), we have
Theorem 2.14
(c.f. Sormani [25]) Suppose \(M^m_i=(X_i, d_i, T_i)\) are integral current spaces which converge in the intrinsic flat sense to a nonzero integral current space \(M^m_\infty =(X_\infty , d_\infty , T_\infty )\). Suppose there exists \(r_0>0\) and a sequence \(p_i \in M_i\) such that for almost every \(r\in (0, r_0)\) we have integral current spaces, \(S(p_i,r)\), for all \(i\in {\mathbb {N}}\) and
Then there exists a subsequence, also denoted \(M_i\), such that \(p_{i}\) converges to \(p_\infty \in {\bar{X}}_\infty \).
Theorem 2.15
(c.f. Sormani [25]) Let \(M_i=(X_i, d_i, T_i)\) and \(M'_i=(X'_i,d'_i,T_i)\) be integral current spaces with
such that
Fix \(\delta >0\). Let \(F_i: M_i \rightarrow M'_i\) be continuous maps which are isometries on balls of radius \(\delta \):
Then, when \(M_\infty \ne \mathbf{{0}}\), we have \(M'_\infty \ne \mathbf{{0}}\) and there is a subsequence, also denoted \(F_i\), which converges to a (surjective) local current preserving isometry.
More specifically, there exists distance preserving maps of the subsequence \(\varphi _i: X_i \rightarrow Z, \varphi '_i: X'_i \rightarrow Z'\), such that
and for any sequence \(p_i\in X_i\) converging to \(p\in X_\infty \):
we have
When \(M_\infty =\mathbf{{0}}\) and \(F_i\) are surjective, we have \(M'_\infty =\mathbf{{0}}\).
3 Sewing Riemannian Manifolds with Positive Scalar Curvature
The main technique we will introduce in this paper is the construction of three-dimensional manifolds with positive scalar curvature through a process we call “sewing” which involved gluing a sequence of tunnels along a curve. We apply Lemma 2.1 which constructs Gromov–Lawson Schoen–Yau tunnels. The lemma is proven in the Appendix.
3.1 Gluing Tunnels Between Spheres
We begin by gluing tunnels between arbitrary collections of pairs of spheres as in Fig. 2.
Proposition 3.1
Given a complete Riemannian manifold, \(M^3\), and \(A_{0} \subset M^3\) a compact subset with an even number of points \(p_{i} \in A_{0}\), \(i = 1, \ldots , n\), with pairwise disjoint contractible balls \(B(p_i,\delta )\) which have constant positive sectional curvature K, for some \(\delta >0\), define \(A_{\delta } = T_{\delta }(A_{0})\) and
where \(U_i\) are the tunnels as in Lemma 2.1 connecting \(\partial B(p_{2j+1},\delta /2)\) to \(\partial B(p_{2j+2},\delta /2)\) for \(j=0,1,\ldots ,n/2-1\). Then given any \(\epsilon >0\), shrinking \(\delta \) further, if necessary, we may create a new complete Riemannian manifold, \(N^3\),
satisfying
and
If, in addition, \(M^3\) has nonnegative or positive scalar curvature, then so does \(N^3\). In fact,
If \(\partial M^3 \ne \emptyset \), the balls avoid the boundary and \(\partial M^3\) is isometric to \(\partial N^3\).
Definition 3.2
We say that we have glued the manifold to itself with a tunnel between the collection of pairs of sphere \(\partial B(p_i,\delta )\) to \(\partial B(p_{i+1},\delta )\) for \(i=1\) to \(n-1\). See Fig. 2.
Gluing two spheres with a tunnel
Proof
For simplicity of notation, set \(A=A_{\delta }\) and \(A'=A_{\delta }'\).
By induction on n and Lemma 2.1, we see that \(N^{3}\) can be given a metric of positive scalar curvature whenever \(M^{3}\) has positive scalar curvature.
Using the fact that the balls are pairwise disjoint and of the same volume, and (10) from Lemma 2.1, we have the volume of \(A'\) can be estimated:
which yields the right-hand side of (62).
Similarly,
which yields the left-hand side of (62).
To estimate the volume of N we will use the volume estimates for \(A'\). Using (10) from Lemma 2.1 again, we have
which yields the right-hand side of (63).
Similarly,
which yields the left-hand side of (63).
Finally, observe that (64) follows since Lemma 2.1 shows that the tunnels \(U_{i}\) have positive scalar curvature. \(\square \)
3.2 Sewing Along a Curve
We now describe our process we call sewing along a curve, where a sequence of balls is taken to be located along curve much like holes created when stitching a thread. We glue a sequence of tunnels to the boundaries of these balls as in Fig. 3. We say that we have sewn the manifold along the curve C through the given balls. By gluing tunnels in this precise way, we are able to shrink the diameter of the edited tubular neighborhood around the curve because travel along the curve can be conducted efficiently through the tunnels.
Sewing a manifold through eight balls along a curve
Proposition 3.3
Given a complete Riemannian manifold, \(M^3\), and \(A_{0}\subset M^3\) Riemannian isometric to an embedded curve, \(C:[0,1]\rightarrow {\mathbb {S}}^3_K\) possibly with \(C(0)=C(1)\) and parametrized proportional to arclength, in a standard sphere of constant sectional curvature K, define \(A_{a} = T_{a}(A_{0})\) as in Proposition 3.1 and assume that \(A_{a}\) is Riemannian isometric to \(T_{a}(C) \subset {\mathbb {S}}^3_K\). Then, given any \(\epsilon >0\) there exists n sufficiently large and \(\delta =\delta (\epsilon ,n,C,K)>0\) sufficiently small as in (66) so that we can “sew along the curve” to create a new complete Riemannian manifold \(N^3\),
exactly as in Proposition 3.1, for
where h is defined in Lemma 2.1 and the disjoint balls \(B(p_i,\delta )\) are to be centered at
and
Thus, the tunnels \(U_{2j+1}\) connect \(\partial B(p_{2j+1},\delta )\) to \(\partial B(p_{2j+2},\delta )\) for \(j=0,1,\ldots , n-1\).
Furthermore,
and
and
Since
we say we have sewn the curve, \(A_{0}\), arbitrarily short.
If, in addition, \(M^3\) has nonnegative or positive scalar curvature, then so does \(N^3\). In fact,
If \(\partial M^3 \ne \emptyset \), the balls avoid the boundary and \(\partial M^3\) is isometric to \(\partial N^3\).
Proof
By the fact that C is embedded, for n sufficiently large, the balls in the statement are disjoint even when \(C(0)=C(1)\) so we may apply Proposition 3.1 to get (69) and (70).
For simplicity of notation, let \(A=A_{\delta }\) and \(A'=A_{\delta }'\).
We now verify the diameter estimate of \(A'\), (71). To do this, we define sets \(C_i \subset A'\) which correspond to the sets \(\partial B(p_i,\delta /2) \subset A\) which are unchanged because they are the boundaries of the edited regions:
whenever i is an odd value. Let
Let x and y be arbitrary points in \(A'\). We claim that there exists \(j,k \in \{1,\ldots ,2n\}\) such that
By symmetry we need only prove this for x. Note that in case I where
we can view x as a point in A. Let \(\gamma _1 \subset A\) be the shortest path from x to the closest point \(c_x \in C[0,1]\) so that \(L(\gamma _1) < \delta \).
If
then
and we have that (76) holds. Otherwise, still in Case I, if (78) fails then we have
where the last inequality follows from \(d_{A'\setminus U}(x,c_x) \le L(\gamma _1) <\delta \) and the fact that \(c_x \in C([0,1])\) is at most L(C) / (2n) away from the boundary of the nearest tunnel.
Alternatively, we have case II where \(x \in U\). In this case, there exists j such that \(x \in U_{2j+1}\) and so
Thus, we have the claim in (76).
We now proceed to prove (71) by estimating \(d_{A'}(x,y)\) for \(x,y \in A'\). If \(j=k\) in (76), then \(d_{A'}(x,y) \le 2(\delta + L(C)/(2n)+h(\delta ))\) and we are done. Otherwise, by (76) and the triangle inequality, we have
Without loss of generality, we may assume that \(j<k\) and that j is odd. Thus, \(C_j \subset \partial U_j\). If k is also odd then by the triangle inequality
and, when k is even,
We know that \({\text {Diam}}(U_j) = \cdots ={\text {Diam}}(U_k) \le h(\delta )\) from (7) of Lemma 2.1, and that the distance between any two adjacent tunnels is the same, and that there are at most n tunnels. Thus, in either case (85) or (86) we have
and by construction the distance between adjacent tunnels is
since the balls \(B(p_i,\delta /2)\) have constant sectional curvature K.
Therefore, combining (84), (87), and (89) we conclude that
which is the desired diameter estimate (71).
We observe that by our choice of \(\delta \) satisfying (66) and the fact that \(h(\delta ) = O(\delta )\) from Lemma 2.1 we have that (72) holds.
Finally, observe that (73) follows since Lemma 2.1 shows that the tunnels \(U_{i}\) have positive scalar curvature. \(\square \)
A two sphere with the equator pulled to a point
4 Pulled String Spaces
The following notion of a pulled string metric space captures the idea that if a metric space is a patch of cloth and a curve in the patch is sewn with a string, then one can pull the string tight, identifying the entire curve as a single point, thus creating a new metric space. This notion was first described to the third author by Burago when they were working ideas related to [6]. See Fig. 4.
Proposition 4.1
The notion of a metric space with a pulled string is a metric space \((Y, d_Y)\) constructed from a metric space \((X,d_X)\) with a curve \(C:[0,1]\rightarrow X\), so that
where for \(x_i \in Y\) we have
and for \(x_i \in X \setminus C[0,1]\) we have
If (X, d, T) is a Riemannian manifold then \((Y,d,\psi _\#T)\) is an integral current space whose mass measure is the Hausdorff measure on Y and
If \((X, d_X, T)\) is an integral current space then \((Y, d_Y, \psi _{\#}T)\) is also an integral current space where \(\psi : X\rightarrow Y\) such that \(\psi (x)=x\) for all \(x\in X\setminus C[0,1]\) and \(\psi (C(t))=p_0\) for all \(t\in [0,1]\). So that
We will in fact prove this proposition as a consequence of two lemmas about spaces with arbitrary compact subsets pulled to a point. Lemma 4.2 proves such a space is a metric space and Lemma 4.3 proves (94) and (95).
4.1 Pulled String Spaces Are Metric Spaces
Lemma 4.2
Given a metric space \((X, d_X)\) and a compact set \(K \subset X\), we may define a new metric space \((Y, d_Y)\) by pulling the set K to a point \(p_0 \in K\) by setting
and, for \(x \in Y\), we have
and, for \(x_i \in Y \setminus \{p_0\}\), we have
Proof
We first prove that \((Y,d_Y)\) is a metric space. By definition, it is easy to see that \(d_Y\) is nonnegative and symmetric. To prove that \(d_Y\) satisfies the axiom of positivity, assume \(x_1=x_2\). Then either \(x_i=p_0\), and \(d_Y(x_1,x_2)=0\) by Definitions (96)–(97), or \(x_i \ne p_0\) and \(d_X(x_1,x_2)=0\) so by (98) we have \(d_Y(x_1,x_2) \le d_X(x_1,x_2)=0\). Conversely, if \(d_Y(x_1,x_2)=0\) then either \(d_X(x_1,x_2)=0\) or
In the first case, \(x_1=x_2\) since \(d_X\) is a metric, so assume otherwise. Then \(d_X(x_1,x_2) \ne 0\) and (99) holds. Being that (99) is a sum of nonnegative numbers, it follows that \(d_X(x_1,y_1)=0\) and \(d_X(x_2,y_2)=0\) for some \(y_i \in K\). Hence, \(x_i=y_i\) which is impossible by the definition of Y unless \(x_1=x_2=p_0\) which yields a contradiction. This proves that \(d_Y\) satisfies positivity.
Next, let us note that by virtue of (97) and (98), we always have
and
for some \(y_i \in K\).
We now verify the triangle inequality: for any \(x_1, x_2, x_3 \in Y\), we need to prove
It will be convenient to define \(y_i \in K\) such that
Assume in Case I that \(d_Y(x_1,x_2) \ne d_X(x_1,x_2)\). Then by (101) and (103),
We have three possibilities: (i) \(d_Y(x_1,x_3) \ne d_X(x_1,x_3)\) and \(d_Y(x_2,x_3) \ne d_X(x_2,x_3)\); (ii) \(d_Y(x_1,x_3) = d_X(x_1,x_3)\) and \(d_Y(x_2,x_3) = d_X(x_2,x_3)\); and (iii) (without loss of generality) \(d_Y(x_1,x_3) \ne d_X(x_1,x_3)\) and \(d_Y(x_2,x_3) = d_Y(x_2,x_3)\).
In Case I (i), we have
In Case I (ii), we have
In Case I (iii), we have
so that
This proves the triangle inequality, (102), in Case I. Next, we assume, in Case II, that \(d_Y(x_1,x_2) = d_X(x_1,x_2)\).
Again, we have three possibilities: (i) \(d_Y(x_1,x_3) \ne d_X(x_1,x_3)\) and \(d_Y(x_2,x_3) \ne d_X(x_2,x_3)\); (ii) \(d_Y(x_1,x_3) = d_X(x_1,x_3)\) and \(d_Y(x_2,x_3) = d_X(x_2,x_3)\); and (iii) (without loss of generality) \(d_Y(x_1,x_3) \ne d_X(x_1,x_3)\) and \(d_Y(x_2,x_3) = d_Y(x_2,x_3)\).
In Case II (i), we have
In Case II (ii), (102) follows immediately from the triangle inequality for \(d_X\).
Finally, in Case II (iii),
which completes the proof. \(\square \)
4.2 Hausdorff Measures and Masses of Pulled String Spaces
Lemma 4.3
If \((X, d_X, T)\) is an integral current space with a compact subset \(K \subset X\) then \((Y, d_Y, \psi _{\#}T)\) is also an integral current space where \((Y, d_Y)\) is defined as in Lemma 4.2 and where \(\psi : X\rightarrow Y\) such that \(\psi (x)=x\) for all \(x\in X\setminus K\) and \(\psi (q)=p_0\) for all \(q\in K\). In addition
If \((X,d_X,T)\) is a Riemannian manifold then \((Y,d_Y,\psi _\#T)\) is an integral current space whose mass measure is the Hausdorff measure on Y and
Proof
We must show that \((Y,d_Y,\psi _\# T)\) is an integral current space. We first observe that \(\psi \) as defined in the statement of the proposition is a 1-Lipschitz function: for \(x,y \in X\setminus K\), there is no ambiguity so we may view them as elements of \(Y\setminus \{p_0\}\) and \(d_Y(\psi (x),\psi (y))=d_Y(x,y) \le d_X(x,y)\) by definition of \(d_Y\). Otherwise, we may assume, without loss of generality, that \(x \in K\) and \(y \notin K\). In this case, \(d_Y(\psi (x),\psi (y))=d_Y(p_0,\psi (y))=d_Y(p_0,y)=\min \{d_X(z,y) : z \in K\} \le d_X(x,y)\), as \(x\in K\). Thus, \(\psi _\# T\) is an integral current on Y since \(\psi \) is a 1-Lipschitz function and the well-known inequality
implies that \(\psi _\# T\) has finite mass because T does. To show that \((Y,d_Y,\psi _\# T)\) is an integral current space there remains to show that it is completely settled, or \(\psi _\# T\) has positive density at \(p_0\).
Let \(f:Y \rightarrow {\mathbb {R}}\) be a bounded Lipschitz map and \(\pi _j:Y \rightarrow {\mathbb {R}}\) be Lipschitz maps. Then
by locality since \(\pi _i \circ \psi \) are constant on \(\{{1}_K \ne 0\}\) (see [2]) so
So, using the characterization of mass from [2], (2.6) of Proposition 2.7,
because \(\psi (x)=x\) on \(X\setminus K\), so since \({{\mathbf {M}}}(\cdot )=\Vert \cdot \Vert \),
where the supremum is taken over all Borel partitions \(\{A_{j}\}\) of X such that \(X = \cup _j A_j\) and all Lipschitz functions \(\pi _i^j \in {\text {Lip}}(X)\) with \({\text {Lip}}(\pi _i^j) \le 1\), then continuing
where the second supremum is taken over all Borel partitions \(\{\tilde{A}_{j}\}\) of \(X\setminus K\) such that \(X\setminus K = \cup _j \tilde{A}_j\) and all Lipschitz functions \({\tilde{\pi }}_i^j \in {\text {Lip}}(X\setminus K)\) with \({\text {Lip}}({\tilde{\pi }}_i^j) \le 1\). So, by the characterization of mass we have
which proves (107).
Finally, assume that the m-dimensional integral current space \((X,d_X,T)\) is a Riemannian manifold. We show that the mass measure of \((Y,d_Y,\psi _\#T)\) is the Hausdorff measure on \((Y,d_Y)\).
We claim that
First, observe that since \(\psi \) is 1-Lipschitz,
by Proposition 3.1.4 on page 37 from [3], hence
Thus, there remains to show the opposite inequality in (110).
Define sets
for each \(j \in {\mathbb {N}}\). Then the \(C_{j}\) are closed sets, \(C_{j} \subset C_{j+1}\) and \(Y\setminus \{p_{0}\} = \cup _{j \in {\mathbb {N}}} C_{j}\). So we may use Theorem 1.1.18 from [3]:
Consider, for each \(j \in N\),
which are closed in X, \(D_{j} \subset D_{j+1}\), and \(X \setminus K = \cup _{j \in {\mathbb {N}}} D_{j}\). Using Theorem 1.1.8 from [3] again:
Next, we claim that
Fix j. Fix \(\delta < \frac{1}{2j}\). Let \(\{ E_{l} \}_{l \in {\mathbb {N}}}\) be a countable cover of \(C_{j}\) with \({\text {Diam}}(E_{l})<\delta \), for all l. Then
To see this, assume otherwise. Then since \({\text {dist}}_{Y}(p_{0},E_{l})<\frac{1}{2j}\) and the definition of distance (as an infimum), there is \(e \in E_{l}\) such that \(d_{Y}(p_{0},e)<\frac{1}{2j}\). Now, we also know that \(E_{l} \cap C_{j} \ne \emptyset \). So, there is \(c \in C_{j} \cap E_{l}\). So, \(d_{Y}(e,c) \le {\text {Diam}}_{Y}(E_{l})< \delta < \frac{1}{2j}\). Also, by the triangle inequality, \(d_{Y}(p_{0},c) \le d_{Y}(p_{0},e)+d_{Y}(e,c) < 1/j\). But this contradicts that \(c \in C_{j}\) as by definition of \(C_{j}\), \(d_{Y}(p_{0},c)>1/j\).
Next, we show that
i.e., \(\psi ^{-1}\) is an isometry when restricted to \(\{E_{l}\}\). In fact, we prove
Let \(a,b \in E_{l}\). Then since \({\text {Diam}}(E_{l})<\delta <\frac{1}{2j}\) we have \(d_{Y}(a,b) \le {\text {Diam}}_{Y}(E_{l})< \delta < \frac{1}{2j}\), so
By definition of the distance \(d_{Y}\), since \(\psi ^{-1}(a)=a\) and \(\psi ^{-1}(b)=b\),
If \(d_{Y}(a,b)=d_{X}(a,b)\), we’re done. If not, then there exists \(k_{1},k_{2} \in K\) so that
By (114),
which implies
But then
which is a contradiction.
Next, observe that \(\{\psi ^{-1}(E_{l})\}_{l \in {\mathbb {N}}}\) is necessarily a cover of \(D_{j}\) so
Taking the infimum over all covers of \(C_{j}\) with diameters less than \(\delta \) gives
then taking the limit as \(\delta \rightarrow 0\) shows
which proves the claim (113).
To finish, we take the limit in (113) as \(j \rightarrow \infty \) and use (111) and (112) to complete the proof. \(\square \)
5 Sewn Manifolds Converging to Pulled Strings
In this section, we consider sequences of sewn manifolds being sewn increasingly tightly and prove they converge in the Gromov–Hausdorff and Intrinsic Flat sense to metric spaces with pulled strings.
To be more precise, we consider the following sequences of increasingly tightly sewn manifolds:
Definition 5.1
Given a single Riemannian manifold, \(M^3\), with a curve, \(A_0=C([0,1])\subset M\), with a tubular neighborhood \(A=T_a(A_0)\) which is Riemannian isometric to a tubular neighborhood of a compact set \(V \subset {\mathbb {S}}^3_K\), in a standard sphere of constant sectional curvature K, satisfying the hypothesis of Proposition 3.3. We can construct its sequence of increasingly tightly sewn manifolds, \(N_j^3\), by applying Proposition 3.3 taking \(\epsilon =\epsilon _j \rightarrow 0\), \(n=n_j \rightarrow \infty \), and \(\delta =\delta _j\rightarrow 0\) to create each sewn manifold, \(N^3=N_j^3\) and the edited regions \(A_{\delta }'=A_{\delta _{j}}'\) which we simply denote by \(A_{j}'\). This is depicted in Fig. 5. Since these sequences \(N_j^3\) are created using Proposition 3.3, they have positive scalar curvature whenever \(M^3\) has positive scalar curvature, and \(\partial N_j^3=\partial M^3\) whenever \(M^3\) has a nonempty boundary.
A sequence of increasingly tightly sewn manifolds
In this section, we prove Lemmas 5.5, 5.6, and 5.7 which immediately imply the following theorem:
Theorem 5.2
The sequence \(N_j^3\) as in Definition 5.1 converges in the Gromov–Hausdorff sense
the metric measure sense
and the intrinsic flat sense
where \(N_\infty \) is the metric space created by pulling the string, \(A_0=C([0,1])\subset M\), to a point as in Proposition 4.1.
In fact, our lemmas concern more general sequences of manifolds which are constructed from a given manifold M and scrunch a given compact set \(K\subset M\) down to a point as follows:
Definition 5.3
Given a single Riemannian manifold, \(M^3\), with a compact set, \(A_0\subset M\). A sequence of manifolds,
is said to scrunch \(A_0\) down to a point if \(A_{\delta }=T_{\delta }(A_0)\) and \(A'_\delta \) satisfies:
and
and
where \(\epsilon =\epsilon _j \rightarrow 0\) and where \(H=H_j \rightarrow 0\) and \(2\delta _j<H_j\).
Note that by Proposition 3.3, a sequence of increasingly tightly sewn manifolds sewn along a curve C([0, 1]) as in Definition 5.1 is a sequence of manifolds which scrunches \(A_0=C([0,1])\) down to a point as in Definition 5.3. So we will prove lemmas about sequences of manifolds which scrunch a compact set and then apply them to prove Theorem 5.2 in the final subsection of this section.
5.1 Constructing Surjective Maps to the Limit Spaces
Before we prove convergence of the scrunched sequence of manifolds to the pulled thread space, we construct surjective maps from the sequence to the proposed limit space.
Lemma 5.4
Given \(M^3\) a compact Riemannian manifold (possibly with boundary) and a smooth embedded compact zero to three-dimensional submanifold \(A_0\subset M^3\) (possibly with boundary), and \(N_j\) as in Definition 5.3. Then for j sufficiently large there exist surjective Lipschitz maps
where \(N_\infty \) is the metric space created by taking \(M^3\) and pulling \(A_0\) to a point \(p_0\) as in Lemmas 4.2–4.3.
Note that when \(A_0\) is the image of a curve, \(N_\infty \), is a pulled thread space as in Proposition 4.1.
Proof
First observe that by the construction in Definition 5.3 there are maps
which are Riemannian isometries on regions which avoid \(A_0\) and map \(A_0\) to \(p_0\). These define Riemannian isometries
In addition, sufficiently small balls lying in these regions are isometric to convex balls in \(M^3\).
Observe also that for \(\delta >0\) sufficiently small, the exponential map:
is invertible where
Taking \(\delta =\delta _{A_0}>0\) even smaller (depending on the submanifold \(A_0\)), we can guarantee that \(\forall v_i\in V_p, |v_i|<2\delta _{A_0}, t_i\in (0,1)\) we have
This is not true unless \(A_0\) is a smooth embedded compact submanifold with either no boundary or a smooth boundary.
Define \(F_j: N_j^3 \rightarrow N_\infty \) as follows:
and
Between these two regions, we take
where \(f_j: N_\infty \rightarrow N_\infty \) is a surjective map:
which takes a point q to
where \(\gamma _q\) is the unique minimal geodesic from \(\gamma _q(0)=p_0\) to \(\gamma _q(1)=q\). Here we are assuming \(\delta _j<\delta _{A_0}\). So
and
In particular for \(x\in \partial T_{\delta _j}(A_j')\),
and for \(x\in \partial A_j'\),
so that \(F_j\) is continuous.
We claim
Only the middle part is difficult. By the definition of \(d_{N_\infty }\), we have the following two possibilities
In Case II, we see that the minimal geodesic from \(q_1\) to \(q_2\) passes through \(p_0\). Since \(f_j(q_1)\) and \(f_j(q_2)\) lie on this geodesic, we have
In Case I, we apply (130) with
because \(t_{i} \in (0,1)\) due to (141) so that by the reverse triangle inequality
to see that
This gives our claim.
We claim \({\text {Lip}}(F_j)\le 4\) everywhere. Given \(x_1,x_2\in N_j^3\), we have a minimizing geodesic \(\eta :[0,1]\rightarrow N_j\) such that \(\eta (0)=x_1\) and \(\eta (1)=x_2\). Then
Since \(|(F_j\circ \eta )'(t)| \le 2|\eta '(t)|\) by our localized Lipschitz estimates and because the function \(F_j\) is continuous, we are done. \(\square \)
5.2 Constructing Almost Isometries
See Sect. 2.2 for a review of the Gromov–Hausdorff distance.
Lemma 5.5
Given \(N_j^{3}\) as in Definition 5.3, the maps \(F_j: N_j^3 \rightarrow N_\infty \) defined in (131)–(133) in the proof of Lemma 5.4 are \(H_j\)-almost isometries with \(\lim _{j\rightarrow \infty }H_j=0\). Thus
Proof
Before we begin the proof recall that
By Theorem 2.3 of Gromov, to prove (155) it suffices to show that \(F_j\) are \(H_j\)-almost isometries. To see this, examine \(x,y\in N_j\) and join them by a minimizing curve \(\sigma :[0,1]\rightarrow N_j\).
If \(\sigma [0,1]\subset N_j \setminus A'_j\), then by (131) we have
and so
Otherwise we have
Next we join \(F_j(x)\) to \(F_j(y)\) by a minimizing curve \(\gamma \). If \(\gamma [0,1]\subset N_\infty \setminus B_{\delta _j}(p_0)\) then there is a curve \(\eta \) such that \(\gamma =F_j\circ \eta \) with \(\eta [0,1]\subset N_j \setminus A'_j\) and so by (131)
Otherwise we have
Hence, \(F_j\) is an \(H_j\) isometry since \(2\delta _{j}<H_{j}\). \(\square \)
5.3 Metric Measure Convergence
Recall metric measure convergence as reviewed in Sect. 2.3.
Lemma 5.6
Given \(N_j^3 \rightarrow N_\infty \) as in Lemma 5.4 endowed with the Hausdorff measures, then we have metric measure convergence if \(A_0\) has \({\mathcal {H}}^{3}\)-measure 0.
Proof
Recall the maps \(F_j: N_j^3 \rightarrow N_\infty \) defined in (131)–(133) in the proof of Lemma 5.4. We need only show that for almost every \(p\in N_\infty \) and for almost every \(r<r_p\) sufficiently small we have
where \(F_j(p_j)=p\) and that for any sequence \(p_{0j}\rightarrow p_0\) we have \(r_0\) sufficiently small that for all \(r<r_0\)
In fact, take any \(p\ne p_0\) in \(N_{\infty }\) and choose
Then for j large enough that \(\delta _j< r_p\) we have
Thus
Thus by (131), \(F_j\) is an isometry from \(B(p_j,r) \subset N_j^3\) onto \(B(p,r)\subset N_\infty \) and so we have
Next we examine \(p_0\). Observe that by (108)
For any \(p_{0,j}\rightarrow p_0\), we have by (125)
Thus
So
Thus
since we claim that
This follows because \(\epsilon _{j} \rightarrow 0\) and (122) implies
The assumption that \({\mathcal {H}}^{3}(A_{0})=0\) then implies (182) after taking the limit.
Similarly, we have for j sufficiently large
So
Thus
which completes the proof. \(\square \)
5.4 Intrinsic Flat Convergence
For a review of intrinsic flat convergence see Sect. 2.5.
Lemma 5.7
Let \(N_j^3 {\mathop {\longrightarrow }\limits ^{ \text {GH}}} N_\infty \) be exactly as in Lemmas 5.4 and 5.5 where we assume M is compact and we have a compact set, \(A_0\subset M\setminus \partial M\). Then there exists an integral current space N such that \({\bar{N}}\) is isometric to \(N_\infty \) and
and when \(A_0\) has Hausdorff measure 0
When \(A_0=C([0,1])\) then \(N=N_\infty \).
Proof
By (123), we have uniformly bounded volume
Since \(\partial N_j^3=\partial M^3\), we have uniformly bounded boundary volume
Combining this with Lemma 5.5 and Theorem 2.6, there exists an integral current space N possibly \(N=\mathbf{{0}}\) such that a subsequence
We claim that \(N\ne \mathbf{{0}}\). If not, then by the final line in Lemma 2.12, for any sequence \(p_j\in N_j\) and almost every r, \(S(p_j,r) {\mathop {\longrightarrow }\limits ^{{\mathcal {F}}}} \mathbf{{0}}\). However, taking \(p_j\) and r such that
we know there is some \(p \in M^3\) with \(B(p,r) \subset N_\infty \setminus \{p_0\}\) that \(d_{{\mathcal {F}}}(S(p_j,r), S(p,r))=0\) for \(p\in M^3\), so \(S(p_j,r) {\mathop {\longrightarrow }\limits ^{{\mathcal {F}}}} S(p,r) \ne \mathbf{{0}}\) which is a contradiction.
By Theorem 2.13, we know that after possibly taking a subsequence we obtain a limit map
We claim that \(F_\infty \) is distance preserving. Let \(p,q\in N\). By Theorem 2.11, we have \(p_j,q_j \in N_j\) converging to p, q in the sense of Definition 2.9, i.e.,
Since the \(F_j\) are \(\epsilon _j\)-almost isometries and \(\epsilon _j \rightarrow 0\), we have
By the definition of \(F_\infty \) we have \(F_j(p_j)\rightarrow F_\infty (p)\) and \(F_j(q_j)\rightarrow F_\infty (q)\). Thus
We claim that \(F_\infty \) maps onto at least \(N_\infty \setminus \{p_0\}\). Let \(x \in N_\infty \setminus \{p_0\}\). Since \(F_j\) are surjective, there exists \(x_j\in N_j\) such that \(F_j(x_j)=x\). Since \(x\ne p_0\), we may define
where \(\text {ConvexRad}_M(x)\) is the convexity radius about x viewed as a point in M. Then there exists j sufficiently large such that \(\delta _j<r\) so that
Furthermore, these balls are isometric to the convex ball \(B(x,r)\subset M^3\).
So
Thus by Theorem 2.14 with \(h_0=d_{{\mathcal {F}}}(S(x,r), \mathbf{{0}})\), and \(N_j {\mathop {\longrightarrow }\limits ^{{\mathcal {F}}}} N\), a subsequence of the \(x_j\) converges to \(x_\infty \in N\). By the definition of \(F_\infty \), we have \(F_j(x_j) \rightarrow F_\infty (x_\infty ) \in N_\infty \). But since \(F_j(x_j)=x\) it follows that \(F_\infty (x_\infty )=x\), hence \(F_\infty \) maps onto \(N_\infty \setminus p_0\).
Taking the metric completions of N and \(N_\infty \setminus \{p_0\}\), we have an isometry
Since \(N_j\) are Riemannian manifolds,
By the lower semicontinuity of mass and the metric measure convergence of \(N_j\) to N we know that
On the other hand by (29)
because almost every tangent cone is Euclidean and it has integer weight everywhere. Thus we have (191). In fact, equality in these inequalities implies that N has weight one everywhere.
Recall that the set of an integral current space only includes points of positive density. Since
Thus N is isometric to \(N_\infty \) when this liminf is positive and N is isometric to \(N_\infty \setminus \{p_0\}\) when this liminf is 0. When \(A_0=C([0,1])\) is a curve in a 3-dimensional Riemannian manifold we have
Thus N is isometric to \(N_\infty \).
Thus N does not depend on the subsequence in (194) and in fact the original sequence (given a consistent orientation) converges in the intrinsic flat sense to N. \(\square \)
5.5 The Proof of Theorem 5.2
Proof
In Proposition 3.3, we show that given any \(\epsilon _j \rightarrow 0\) we can find \(n_j \rightarrow \infty \) and \(\delta _j \rightarrow 0\) so fast that \(\delta _jn_j \rightarrow 0\) and we have \(h(\delta _j)n_j \rightarrow 0\) as well such that the sewn manifolds:
satisfy:
and
and
where
Thus we have a sequence \(N_j\) which is scrunching a set \(A_0=C([0,1])\) to a point as in Definition 5.3.
Lemma 5.5 implies that
where \(N_\infty \) is the pulled string space. Lemma 5.6 implies we have metric measure to \(N_\infty \) convergence because \(A_0=C([0,1])\) has \({\mathcal {H}}^{3}\)-measure 0.
Lemma 5.7 implies that
and
completing the proof of Theorem 5.2. \(\square \)
6 Sewing a Sphere to Obtain our Limit Space
Here we construct the specific example of a sequence of manifolds with positive scalar curvature that converges to a limit space which fails to have generalized nonnegative scalar curvature as discussed in the introduction. More specifically:
Example 6.1
We define a sequence \(N_j^3\) of manifolds with positive scalar curvature constructed from the standard \({\mathbb {S}}^3\) sewn along a closed geodesic \(C:[0,1]\rightarrow {\mathbb {S}}^3\) with \(\delta =\delta _j \rightarrow 0\) as in Proposition 3.3. Then by Theorem 5.2 we have
where \(N_\infty \) is the metric space created by taking the standard sphere and pulling the geodesic to a point as in Proposition 4.1. By Lemma 6.3 below we see that at the pulled point \(p_0\in N_\infty \), we have (3). Thus we have produces a sequence of three-dimensional manifolds with positive scalar curvature converging to a limit space which fails to satisfy generalized scalar curvature defined using limits of volumes of balls as in (1).
Remark 6.2
Note that with \(\delta _j \rightarrow 0\), the neck in the center of the tunnels has a rotationally symmetric minimal surface whose area is \(\le 4\pi \delta _j^2\) which converges to 0. So this sequence, and in fact any sewn sequence created as in Definition 5.1, has \(\text {MinA}(N_j) \rightarrow 0\).
Lemma 6.3
At the pulled point \(p_0\in N_\infty \) of Example 6.1 we have
Proof
First, observe that
Since C([0, 1]) is a closed geodesic of length \(2\pi \) in a three-dimensional sphere, we have
Thus
as claimed. \(\square \)
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Acknowledgements
J. Basilio was partially supported as a doctoral student by NSF DMS 1006059. C. Sormani was partially supported by NSF DMS 1006059.
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J. Basilio was partially supported as a doctoral student by NSF DMS 1006059. C. Sormani was partially supported by NSF DMS 1006059.
Appendix: Short Tunnels with Positive Scalar Curvature by Jorge Basilio and Józef Dodziuk
Appendix: Short Tunnels with Positive Scalar Curvature by Jorge Basilio and Józef Dodziuk
There is a deep connection between the geometry of Riemannian manifolds \(M^n\) with positive scalar curvature and surgery theory. The subject began with the surprising discovery by Gromov and Lawson [12] (for \(n \ge 3\)) and Schoen and Yau [32] that a manifold obtained via a surgery of codimension 3 from a manifold \(M^n\) with a metric of positive scalar curvature may also be given a metric with positive scalar curvature. The key to the tunnel construction of [12] is defining a curve \(\gamma \) which begins along the vertical axis then bends upwards as it moves to the right and ends with a horizontal line segment, cf. Fig. 6 below. The tunnel then is the surface of revolution determined by \(\gamma \). We note that the “bending argument” has attracted some attention (See [24]).
As the goals of the surgery theory were topological in nature, Gromov and Lawson did not estimate with diameters or volumes of these tunnels. Indeed, the tunnels they constructed may be thin but long (See [11]). To build sewn manifolds, we need tunnels with diameters shrinking to zero as the size of the original balls decreases to zero (see (7), (8) (9)). Therefore, we prove Lemma 2.1 to obtain a refinement of the Gromov and Lawson construction showing the existence of tiny (in sense of (10)) and arbitrarily short tunnels with a metric of positive scalar curvature.
Proof of Lemma 2.1
To aid the reader, we provide a summary of our proof and introduce additional notation.
1.1 Outline of Proof of Lemma 2.1
To aid the reader, we provide a summary of our proof and introduce additional notation.
Step 1: Setup and notation
Let \(\epsilon >0\) be given. We shall specify \(0<\delta _0<\delta /2\) below.
Given that \(B_1=B(p_1,\delta /2) \subset M^3\) has constant sectional curvature \(K>0\), we may choose coordinates so that it is realized as a hypersurface of revolution. This is also true for \(B(p_1,\delta _0) \subset B_1\) for \(0<\delta _0<\delta /2\) centered at the same \(p_1\). Thus, \(B(p_1,\delta _0)\) is a hypersurface of revolution \(U_{\gamma _0}'\) with the induced metric in \({\mathbb {R}}^4\) determined by revolving a segment of the circle \(\gamma _0\) in the \((x_0,x_1)\)-plane about the \(x_0\)-axis. We set things up so that the vertical \(x_1\)-axis corresponds to boundary points of \(B(p_1,\delta _0)\). We then proceed as Gromov and Lawson to deform \(\gamma _0\) away from vertical axis bending it upwards as we move to the right and ending with an arbitrarily short horizontal line segment. We call this curve \(\gamma \), cf. Fig. 6. The curve \(\gamma \) begins exactly as \(\gamma _0\) so that we may attach the corresponding hypersurface onto the larger \(B(p_1,\delta /2)\) in a natural way. We do exactly the same for \(B_2 \subset M^3\) and identify the two hypersurfaces along their common boundary, i.e., the “tiny neck,” forming \(2U_{\gamma }'=U_{\gamma }' \sqcup U_{\gamma }'\). We then define the tunnel \(U=U_\delta \) by
where \(0<\delta _0<\delta /2\) and \(U_{\gamma }'=U_{\delta _0,\gamma }'\) is a modified Gromov–Lawson tunnel, see Fig. 1.
The boundary of \(2U_{\gamma }'\) is isometric to a collar of \(B(p_1,\delta _0) \sqcup B(p_2,\delta _0)\), so we may smoothly attach it to form (224).
Step 2: Construction of the curve\(\gamma \), Part 1:\(C^{1}\)
In this step, we construct a \(C^1\), and piecewise \(C^\infty \), curve \(\gamma \). The construction is based on the bending argument of Gromov and Lawson and uses the fundamental theorem of plane curves, i.e., the fact that a smooth curve parametrized by arclength is uniquely determined by its curvature, the initial point, and the initial tangent vector. Care must be taken to ensure that the induced metric on \(U_{\gamma }'\) maintains positive scalar curvature and that the length of \(\gamma \) is controlled to yield diameter and volume estimates of Lemma 2.1. This step is quite technical and forms the heart of the proof.
Step 3: Construction of the curve\(\gamma \), Part 2: from\(C^{1}\)to\(C^{\infty }\)
In this step, we show how to modify the curve constructed in Step 2 to obtain a smooth curve \({\bar{\gamma }}\) while maintaining all the required features. The modification is elementary and, once it is completed, we rename \({\bar{\gamma }}\) back to \(\gamma \).
Step 4: Diameter estimates (7), (9) and volume estimates (10), (11)
This is very straightforward since the previous steps give an estimate of the length of the tunnel.
The curve \(\gamma \)
We remark here that the choice of \(\delta _0\) is used only to insure that the tunnel \(U'\) (see Fig. 1) has sufficiently small volume.
1.2 Step 1 of the Proof
We now set up our notation further, describe U explicitly in terms of a special curve \(\gamma \), and state the important curvature formulas needed in later steps. The construction of \(\gamma \) is done in the next two sub-sections (Steps 2 and 3).
As mentioned in Sect. 1, because we assume that \(B_1\) and \(B_2\) have constant sectional curvature K, we may work directly in Euclidean space \({\mathbb {R}}^4\) with coordinates \((x_0,x_1,x_2,x_3)\) and its standard metric. Let \(\gamma (s)\) be a curve in the \((x_0,x_1)\)-plane, parametrized by arc-length, written as \(\gamma (s)=(x_0(s),x_1(s))\). This curve specifies a hypersurface in \({\mathbb {R}}^4\) (by rotating \(\gamma \) about the \(x_0\)-axis),
which we endow with the induced metric. Our curve \(\gamma \) will always lie in the first quadrant of \((x_0,x_1)\)-plane and will be parametrized so that \(x_0(s)\) will be increasing. We denote by \(\theta (s)\) the angle between the horizontal direction and the upward normal vector, and by \(\varphi (s)\) the angle between the horizontal direction and the tangent vector to \(\gamma \).
We remark that the two angle functions are related by
See Fig. 6. In particular, \(\varphi \in (-\pi /2,0]\).
Denote by k(s) the geodesic curvature of \(\gamma \). It is a signed quantity so that \(\gamma \) bends away from the horizontal axis if \(k(s)>0\) and towards the \(x_0\)-axis when \(k(s)<0\). If \(\gamma (s_0)=(c,d)\) and \(\varphi _0= \varphi (s_0)\) then (cf. Theorem 6.7, [13]) the function k(s) determines \(\gamma \) by the formulae
and
Our aim is to define a function k(s) so that the resulting threefold of revolution \(U'\) has positive scalar curvature. The formula on page 226 of [12] for \(n=3\) gives a relation between the two curvatures. Namely
where \(\mathrm{Scal}_{U'}(s)\) is the scalar curvature of the induced metric on \(U'\) and k is the geodesic curvature of \(\gamma \). In particular, the formula holds if \(\gamma \) is the intersection of the 3-sphere around the origin with the \((x_0,x_1)\)-plane in which case k is a negative constant.
We begin defining our curve \(\gamma (s)\) so that \(\gamma (0)\) corresponds to a point on \(\partial B(p_1,\delta _0)\) and \(\gamma (s)\), for small values of \(s\in [0,s_0]\), parametrizes the intersection of \(B(p_1,\delta _0)\) with the \((x_0,x_1)\)-plane. In particular, for small s, \(k(s)\equiv -\sqrt{K}\). We choose \(s_0 =\delta _0/2\) and then extend (in Step 2, Sect. 1) the function k(s) to a suitable step function on a longer interval [0, L] so that the resulting curve \(\gamma (s)\) has the following properties.
-
(I)
The graph of \(\gamma \) lies strictly in the first quadrant, beginning at \(p_I=\gamma (0)=(0,\cos (-\pi /2+\delta _0)/\sqrt{K})\) and ending at \(p_F=\gamma (L)\) with \(x_0(L)>0\), \(x_1(L)>0\), where L is the length of the curve. Moreover, a point of \(\gamma \) moves to the right when s increases.
-
(II)
Let \(\theta (s)\) be the angle between the upward pointing normal to \(\gamma \) and the \(x_0\)-axis. The curve \(\gamma \) ends at \(p_F\) with \(\theta (L)=\pi /2\) and has \(\theta =\pi /2\) (so that it is a horizontal line segment) for an arbitrarily small interval \((L',L]\) (where \(L'<L\)).
-
(III)
The curve \(\gamma \) has constant curvature \(-\sqrt{K}\) near 0 so that the boundary of U has a neighborhood that is isometric to a collar of \(B_1 \cup B_2\).
-
(IV)
The curvature function k(s) satisfies
$$\begin{aligned} k(s) < \frac{\sin (\theta (s))}{2x_{1}(s)} \qquad s \in [0,L], \end{aligned}$$(230)so that the expression on the right-hand side of (229) is positive for all \(s \in [0,L]\). We remark here that in certain stages of the construction k(s) will have discontinuities so that \(\mathrm{Scal}_{U'}(s)\) is not defined but this will cause no difficulties.
-
(V)
The length of \(\gamma \), L, is \(O(\delta _{0})\).
Due to properties (I) and (II) of \(\gamma \) above, we may smoothly attach two copies of \(U'\) along their common boundary at \(s=L\) to define \(2U'=U_{\gamma }' \sqcup U_{\gamma }'\) and then, using property (III), attach \(2U'\) to form U as in (224).
In the next step, we construct a piecewise \(C^1\) curve \(\gamma \) in the \((x_0,x_1)\)-plane which satisfies properties (I) through (V). Then, in Step 3, we modify the construction once more to produce a smooth curve, \({\bar{\gamma }}\), with these same properties.
1.3 Step 2 of the Proof: Construction of \(\gamma \), Part 1: \(C^{1}\)
As above, let \(s_0=\delta _0/2\) and let \(q_{0}=(a_{0},b_{0})\) be the coordinates of the point \(\gamma (s_{0})\) that is already defined. By choosing \(\delta _0\) sufficiently small, we can assume that the tangent vector to \(\gamma \) at \(s=s_{0}\) is nearly vertical and is pointing downward at \(s=s_{0}\). We also have \(k(s)\equiv -\sqrt{K}\) on \([0,s_0]\).
We will use a finite induction to define a sequence of extensions of \(\gamma \) over intervals \([s_{i},s_{i+1}]\), with \(s_{i}<s_{i+1}\) for a finite number of steps \(0 \le i \le n\), where \(n=n(\delta _{0})\) is the number of steps required such that properties (I), (III), (IV), and (V) all hold at each extension. We denote by \((a_{i},b_{i})\) the coordinates of the point \(\gamma (s_{i})\) for \(0 \le i \le n\).
Let us first choose the curvature function k(s) of \(\gamma (s)\) on the first extended interval \([s_{0},s_{1}]\). Observe that equation (230) limits the amount of positive curvature allowed for k(s). In fact, we choose k(s) to be the constant \(k_{1}>0\) over the interval \([s_{0},s_{1}]\) based only the initial data at \(s_{0}\)
where \(\theta (s_{0})=\frac{\pi }{2}+\varphi (s_{0}) = \delta _{0}-\sqrt{K}s_{0}>0\) and \(b_{0}=x_{1}(s_{0})\). Note that constant positive curvature means that \(\gamma (s)\) moves along the arc of a circle of curvature \(1/\sqrt{k_{1}}\) bending away from the origin.
We verify that property (IV) holds with our choice of \(k_{1}\) in (231). From (227), we see that \(\varphi (s)\) is an increasing function with range in the interval \((-\pi /2,0)\), hence \(\theta (s)\) is also increasing by (226). Moreover, from (227) and (228), we see that the \(x_{1}\)-coordinate function is decreasing on the interval \((s_{0}, s_{1})\) since \(x_{1}'(s)=\sin (\varphi (s))<0\). Thus, the expression on the right-hand side of (230), \(\sin (\theta (s))/(2x_{1}(s))\), is an increasing function on \((s_{0},s_{1})\) so that
Since \(k(s)\equiv k_{1}\) is constant it follows that the property (IV) holds for \(s \in [s_{0},s_{1}]\).
Next, we choose the length of the extension \(\Delta s_{1} = s_{1} -s_{0}\), so that properties (I) and (V) hold. This is achieved by setting
Observe that \(x_0(s)\) is increasing since \(x_{0}'(s)=\cos (\varphi (s))>0\) as \(\varphi \in (-\pi /2,0)\).
Clearly we have
since \(b_0\) is the vertical distance of \(\gamma (s_0)\) to the \(x_0\)-axis which is less than the distance along the sphere.
Of course, we do not achieve a final angle of \(\pi /2\) of the normal at \(s_{1}\) and gain only a small but definite increase in the angle. The change in angle of the normal with the \(x_{0}\)-axis is
With \(\gamma \) extended over the first interval \([s_{0},s_{1}]\), we now inductively define further extensions. Assume that \(\Delta s_{j}\), \(s_{j}\) and \(k_{j}\) have been chosen for \(j=1,2,\ldots ,(i-1)\), and \(\gamma \) extended on the intervals \([s_{j},s_{j+1}]\), we then define
where \(\gamma (s_{i})=(a_{i},b_{i})\). In what follows we will also write \(\theta _j\) and \(\varphi _j\) for \(\theta (s_j)\) and \(\varphi (s_j)\), respectively. We remark that \(b_{i+1} < b_i\) by (228) since the angle \(\varphi \) is negative and that \(k_{i+1} > k_i\) since the ratio \(\frac{\sin (\theta (s))}{x_{1}(s)}\) is increasing. Observe that properties (I), (IV), and (V) of \(\gamma \) hold on \([s_{i-1},s_{i}]\) for all i by our choices in (235) by arguments analogous to those given for the first extension of \(\gamma \) on \([s_{0},s_{1}]\).
We observe that we gain a definite amount of angle \(\theta \) with each extension since, by (235), for each \(j \in \{1,2,\ldots , i\}\),
because \(\theta (s_{j-1}) \ge \theta (s_{0})\) and the values of \(\theta \) are in the range \((0,\pi /2)\) so that the sine is an increasing function. We stop the construction when \(\theta (s)\) reaches the value \(\pi /2\). Thus the total change in the angle \(\theta \) over the interval \([0,s_{i}]\) is bounded from below by
To prove property (V), that the length of \(\gamma \) is on the order of \(\delta _{0}\), we need the sequence of \(b_{i}\)’s to be summable and will want to compare it to the geometric progression. The difficulty here is that, since our curve is bending more and more upwards, the ratios \(b_i/b_{i-1}\) increase. For this reason, we stop our induction when \(\theta \) reaches the value of \(\pi /4\). It will turn out that once this value is reached, we can complete the construction of k(s) by a single extension albeit with \(\Delta s\) not given by (235).
Thus, define \(n=n(\delta _{0})\) to be the first positive integer with
which exists by (237). Moreover, if \(\theta _{n}>\pi /4\) we re-define \(s_{n}\) to be the exact value in \((s_{n-1},\infty )\) such that \(\theta (s_{n})=\pi /4\). Thus, for the modified value of \(s_n\)
The following Lemma gives the desired comparison.
Lemma 7.1
There exists a universal constant \(C\in (0,1)\), independent of \(\delta _{0}\) and K, such that for all \(i \le n\)
where \(n=n(\delta _0)\) is as above.
The Lemma, to be proven shortly below, implies that the length of the curve \(\gamma \) on the entire interval \([0,s_{n}]\) is no larger than a constant (independent of \(\delta _{0}\)) times \(\delta _{0}\). Namely,
Thus, from (235) and Lemma (7.1), we have
by the lemma and (234). So, \(L(\gamma ([0,s_{n}])) \le C_{1} b_{0}\) with \(C_{1}= \frac{1}{2-2C}\) which is independent of \(\delta _{0}\) since C is. This proves that \(L(\gamma ([0,s_{n}])) = O(\delta _0)\).
Proof of Lemma 7.1
Let \(1 \le i \le n\). We compute explicitly using (227), (228), and (235),
and
Thus,
Therefore, by the Mean Value Theorem, there exists \(\mu _{i} \in (\varphi (s_{i-1}),\varphi (s_{i-1})+\sin (\theta (s_{i-1}))/8)\) such that
To complete the proof of the claim, we seek a constant \(0<C<1\), independent of \(\delta _{0}\), such that
Recall that the angle function \(\varphi \) takes negative values throughout.
We claim that the choice
will satisfy our requirement.
This follows from the fact that the sine is an increasing function on the interval \((\varphi (s_{i-1}),\varphi (s_{i-1})+\sin (\theta (s_{i-1}))/8)\) and the fact that both the angles \(\varphi _{i}\) and \(\theta _{i}\) are increasing, so
By our choice of \(s_n\), \(\theta (s_{n})=\pi /4\) from (239) and \(\varphi (s_{n})=-\pi /4\) so that
This finishes the proof of the Lemma. \(\square \)
At this stage of the construction, \(\gamma \) has angle \(\theta =\pi /4\) at the endpoint \(s_{n}\). We make one additional extension of our step function.
We now define \(s_{n+1}>s_{n}\) and \(k_{n+1}>0\) as follows.
By (227) \(\varphi (s)\) in \([s_{n},s_{n+1}]\) will be given by
Let \(s_{n+1}\) be determined by \(k_{n+1}\) as the first value such that \(\varphi (s_{n+1})=0\) (equivalently \(\theta (s_{n+1})=\pi /2\)). Then
so that
We require in addition that \(b(s_{n+1})>0\) (that is, \(\gamma \) remains above the \(x_{0}\)-axis). Using (247) and (228), we obtain
so that \(b(s_{n+1})>0\) is equivalent to
or
On the other hand, \(k_{n+1}\) has to be bounded from above in order to guarantee (230). Therefore, we require that
or
Combining (249) and (250) gives conditions for \(k_{n+1}\)
Since \(\sin (\theta (s))=\cos (\varphi (s))\), (251) is equivalent to
Now, recall that \(s_{n}\) was chosen in (239) so that \(\varphi (s_{n})=-\pi /4\) so
Now, choose arbitrarily any \(\alpha \), satisfying
and define \(k_{n+1}\) by
With this choice (252), and therefore, (249) and (250) hold.
To ensure property (II), we choose \(L>s_{n+1}\) so that \(L-s_{n+1}\) is arbitrarily small. We extend \(\gamma \) to the interval \([s_{n+1},L]\), where \(\gamma \) is a straight horizontal line on \([s_{n+1},L]\) by choosing \(k(s)=0\) there. To check that the length of the curve we constructed is \(O(\gamma _0)\) we observe that
We note that the choice of L is arbitrary. It will be made explicit in the next step when we construct the curve \({\bar{\gamma }}\), the \(C^{\infty }\) version of \(\gamma \).
This completes the construction of the continuously differentiable curve \(\gamma \) defined on the interval [0, L] satisfying properties (I) through (V).
1.4 Step 3 of the Proof: Construction of \(\gamma \), Part 2: From \(C^{1}\) to \(C^{\infty }\)
In this step, barred quantities will refer to the \(C^{\infty }\) curve \({\bar{\gamma }}(s)\) to be constructed in this step and all the other quantities related to the construction (for example, \({\bar{\theta }}\), \({\bar{\varphi }}\), \({\bar{k}}(s)\), etc.). Unbarred quantities will refer to the \(C^{1}\) curve constructed in the previous step.
The general plan is to replace k(s) as chosen in Step 2 with a smooth version \({\bar{k}}(s)\) as depicted in Fig. 7, which will then define \({\bar{\gamma }}\) by the formulae (227) and (228). Set \(k_0=-K^{1/2}\) and modify k(s) on \([s_i,s_{i+1}]\) for \(i=0,1,2,\ldots ,n\) so that the graph of \({\bar{k}}(s)\) will connect to the constant function equal to \(k_{i}\) smoothly at \(s_i\), will rise steeply to the value \(k_{i+1}\) in a very short interval \([s_i,s_i+\alpha ]\) and will connect smoothly with constant function equal to \(k_{i+1}\) in \([s_i+\alpha , s_{i+1}]\). For each \(i=0,1,2,\ldots n\), \({\bar{k}}|[s_i,s_{i+1}]\) can be constructed as follows. Choose and fix a \(C^\infty \) function g(s) which is identically 0 for \(s<0\), identically 1 for \(s>1\), and strictly increasing on [0, 1]. Then \({\bar{k}}| [s_i,s_{i+1}]\) is constructed by appropriate rescaling and translations of the graph of g(s) in both vertical and horizontal directions. The values of \(k_i\) and \(k_{i+1}\) determine the transformations along the vertical axis but rescaling of the independent variable remains a free parameter \(\alpha \) to be set sufficiently small later. We will use the same value of \(\alpha \) for every \(i=1,2,\ldots n\).
Graph of the smooth curvature \({\bar{k}}(s)\) with “full bend”
Graph of the curvature, k(s), with “full bend” as a step function
Since
we loose a small amount of “bend” so that \({{\bar{\theta }}}(s_{n+1}) < \frac{\pi }{2}\) by a very small amount controlled by \(\alpha \). We compensate for this by one final extension of \({\bar{k}}\) to an interval \([s_{n+1},L]\) with \(L=s_{n+1} + 2\beta \). We choose \({\bar{k}}\) so that it connects smoothly with \(k_{n+1}\) at \(s_{n+1}\), drops smoothly to zero over \([s_{n+1}, s_{n+1}+\beta ]\) and continues identically zero on \([s_{n+1}+\beta , s_{n+1}+2\beta ]\). \(\beta \) and \({\bar{k}}\) are chosen so that
This ensures that \({\bar{\theta }}=\frac{\pi }{2}\) in the interval \([s_{n+1}+\beta , s_{n+1}+2\beta ]\). This final extension is constructed as the preceding ones except that we have to use the reflection \(s \mapsto -s\) before rescaling and translating the original function g. We note that \(\beta =O(\alpha )\) is determined by the choice of \(\alpha \) and the requirement that \({\bar{\theta }}(L)=\frac{\pi }{2}\). We also observe that as \(\alpha \) tends to zero, the functions \({\bar{\varphi }}\), \({\bar{\theta }}\), \(\bar{x_0}\), and \(\bar{x_1}\) will converge uniformly on [0, L] to \(\varphi \), \(\theta \), \(x_0\), and \(x_1\), respectively, as follows from (227) and (228).
We now check that the properties (I) through (V) on page (I) hold for the curve \({\bar{\gamma }}\) for sufficiently small choice of \(\alpha \). Only (IV) and (V) need a verification. (V) follows since \(L = s_{n+1} + 2\beta = O(\delta _0) + O(\alpha )\). To prove (IV) we use the uniform convergence on \([0,s_{n+1}]\) as \(\alpha \) approaches 0 of \(\frac{\sin {\bar{\theta }}(s)}{2{\bar{x}}_1(s)}\) to \(\frac{\sin \theta (s)}{2x_1(s)}\). More precisely, on \([s_i,s_{i+1}]\),
For sufficiently small \(\alpha \), the first term on the right becomes positive by the property (IV) for the curve \(\gamma \) while the second term is nonnegative by construction (cf. Fig. 8). Finally, in the last interval \([s_{n+1},L]\) the ratio \(\frac{\sin {\bar{\theta }} (s)}{2{\bar{x}}_1(s)}\) is nondecreasing so that
since the last inequality was verified for \(s=s_{n+1}\) already. Property (IV) follows since \(k_{n+1} > {\bar{k}}(s)\) in \([s_{n+1},L]\). This finishes the construction of \({\bar{\gamma }}\).
1.5 Step 4 of the Proof: Diameter and Volume Estimates of Lemma 2.1
Given the definition of U in (224), the diameter of U is estimated by
To estimate the volume of \(U'\), note that the intersection of \(U'\) with the hyperplane \(x_0=x_0(s)=c\) for \(0<s<L\) is a sphere of two dimensions and of radius \(x_1(s)< \delta _0\). It follows by Fubini’s theorem that \({\text {Vol}}(U')=O(\delta _0^3)\). To prove (10) recall that U is obtained from the union of two disjoint balls of radius \(\delta \) by removing balls of radius \(\delta _0\) and attaching \(U'\) along the common boundary (cf. Fig. 1). Since the volumes of the removed balls and of the added tunnel are \(O(\delta _0^3)\), the estimate (10) follows by choosing \(\delta _0\) sufficiently small depending on \(\epsilon \). The estimate (11) is proved in the same way. The proof of Lemma 2.1 is now complete. \(\square \)
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Basilio, J., Dodziuk, J. & Sormani, C. Sewing Riemannian Manifolds with Positive Scalar Curvature. J Geom Anal 28, 3553–3602 (2018). https://doi.org/10.1007/s12220-017-9969-y
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DOI: https://doi.org/10.1007/s12220-017-9969-y