An intrinsic flat limit of Riemannian manifolds with no geodesics
In this paper we produce a sequence of Riemannian manifolds , , which converge in the intrinsic flat sense to the unit -sphere with the restricted Euclidean distance. This limit space has no geodesics achieving the distances between points, exhibiting previously unknown behavior of intrinsic flat limits. In contrast, any compact Gromov-Hausdorff limit of a sequence of Riemannian manifolds is a geodesic space. Moreover, if , the manifolds may be chosen to have positive scalar curvature.
In 1981, Gromov introduced the Gromov-Hausdorff distance between metric spaces, , and proved that compact Gromov-Hausdorff limits of Riemannian manifolds are geodesic metric spaces [Gro81]. That is, the distance between any pair of points in the limiting metric space is realized by the length of a curve between the points. Recall that by the Hopf-Rinow Theorem, any compact Riemannian manifold with the length metric , defined by taking the infimum of lengths of curves from to , is a geodesic metric space. If the Riemannian manifold is embedded into Euclidean space, one may also study a different metric space, defined using the restricted Euclidean metric . Observe that the standard -dimensional sphere with the Euclidian distance is an example of a metric space with no geodesics. Geodesic spaces and the Gromov-Hausdorff distance are reviewed in Sections 2.1 and 2.3.
In [SW11], the third author and Wenger introduced the intrinsic flat distance between two integral current spaces and . An -dimensional integral current space is a metric space with an collection of oriented biLipschitz charts and an integral -current such that is the set of positive density of . For example, a compact oriented manifold has a canonical integral current space structure, . If is embedded in Euclidean space then we can create a different integral current space, whose metric depends upon that embedding. We review these spaces and in Sections 2.4 and 2.5, respectively.
In the Appendix to [SW11], the second author studied a sequence of Riemannian -spheres , constructed by removing a pair of tiny balls from two standard spheres and connecting the resulting boundaries by an increasingly thin tunnel. These manifolds converge in the Gromov-Hausdorff sense to a metric space, , which is a pair of standard spheres connected by a line segment of length endowed with the length metric, . On the other hand, the intrinsic flat limit of this sequence consists of the pair of spheres without the line segment. See Figure 1.
In other words,
with respect to the intrinsic flat distance. The line segment does not appear in the intrinsic flat limit since the current does not have positive density there. Since is not path-connected, this example shows that intrinsic flat limits of Riemannian manifolds need not be geodesic metric spaces.
When the second author presented this paper at the Geometry Festival in 2009, various mathematicians asked whether it was possible to prove that intrinsic flat limits of Riemannian manifolds are locally geodesic spaces: for all , is there a neighborhood about such that all pairs of points in are joined by a geodesic segment? In this paper we provide an example demonstrating that this is not the case. More precisely, we prove the following theorem.
For each , there is a sequence of closed, oriented -dimensional Riemannian manifolds , so that the corrisponding integral current spaces -converge to
If , the manifolds may be chosen to have positive scalar curvature.
Since the metric space is not a length space, we immediately obtain a negative answer to the above question.
For , the intrinsic flat limit of closed, oriented -dimensional Riemannian manifolds need not be locally geodesically complete.
To construct the Riemannian manifolds in Theorem 1.1, we modify the standard -sphere by gluing in increasingly thin tunnels between pairs in an increasingly dense collection of removed balls. The lengths of the tunnels approximate the Euclidean distance between the points at the centers of the balls they replace. See Figure 2. The original plan for this construction in two dimensions was conceived by the first and last authors and presented a few years ago but never published.
In [Gro14], Gromov suggested that perhaps the natural notion of convergence for sequences of manifolds with positive scalar curvature is the intrinsic flat convergence. Note that the classic Ilmanen Example [SW11] demonstrates that a sequence of three dimensional manifolds of positive scalar curvature with increasingly many increasingly thin increasingly dense wells has no Gromov-Hausdorff limit but does converge naturally in the intrinsic flat sense to a sphere. Thus the second and third authors decided to complete the construction of our example with positive scalar curvature when the dimension is .
In order to achieve the positive scalar curvature condition, we design the tunnels using a refinement of the classical Gromov-Lawson construction [GL80] recently obtained by Dodziuk [Dod18]. See also Schoen-Yau’s classic tunnel construction in [SY79]. Topologically, each is the connected sum of a large number of copies of .
We begin the construction in Section 3 by proving our main technical result, Theorem 3.1, which allows us to estimate the intrinsic flat distance between a space with a tunnel and a space with a thread. This is an extension of the pipe-filling technique developed by the second author in the Appendix of [SW11]. In [SW11], this technique was applied to prove the (the pair of spheres with a tunnel between them) and (the pair of spheres with a thread between them) could be isometrically embedded into a common space. This then allowed one to estimate the intrinsic flat distance and Gromov-Hausdorff distance between and . In Theorem 3.1, we start with an arbitrary Riemannian manifold and a pair of points in that manifold and a length less than the distance between the points. We show that the manifold created by joining the points with a thin enough tunnel of length is close in the intrinsic flat sense to the space created by joining the points with a thread of length [Theorem 3.1]. As before, our estimate on the intrinsic flat distance between the space depends on volumes and diameters while the Gromov-Hausdorff distance depends on the width of the tunnel and diameters.
We apply this pipe filling technique inductively in Proposition 5.1 to prove that our sequence, , of spheres with increasingly dense tunnels [Definitions 4.2 and 4.4] are increasingly close to a sequence, , of spheres with increasingly dense threads [Definitions 4.1 and 4.3]. We apply a direct filling construction to prove that the spheres with the increasingly dense threads viewed as integral current spaces, converge to the sphere with the restricted Euclidean metric, , in the intrinsic flat sense [Proposition 5.2]. Note that the threads are not part of the integral current space, , as they have lower density. These are combined to prove Theorem 1.1.
It should be noted that the manifolds in our sequence have closed minimal hypersurfaces of increasingly small area. These stable minimal surfaces are located within the increasingly thin tunnels. In [Sor17], the third author has conjectured that if one were to impose a uniform positive lower bound on
that this combined with nonnegative scalar curvature would guarantee that the limit space has geodesics between every pair of points. This is a three dimensional conjecture. It would be interesting to explore if other kinds of tunnels might be constructed in higher dimensions to create limits with no geodesics even with a uniform positive lower bound on MinA.
The authors would like to thank Brian Allen, Edward Bryden, Lisandra Hernandez, Jeff Jauregui, Sajjad Lakzian, Dan Lee, Raquel Perales, and Jim Portegies for many conversations about scalar curvature and convergence. We would like to thank Jozef Dodziuk, Misha Gromov, Marcus Khuri, Blaine Lawson, Rick Schoen, and Shing-Tung Yau for their interest in these questions.
Here we provide a brief review of metric spaces, length spaces and integral current spaces, followed by a review of the Gromov-Hausdorff and intrinsic flat distances between these spaces.
2.1. Review of Geodesic Spaces:
Given any metric space, , one may define the length of a rectifiable curve as follows:
If every pair of points in is connected by a rectifiable curve, then one can define the induced length metric, , on as follows:
If is compact, then this infimum is achieved and any curve achieving the infimum is called a minimizing geodesic. A geodesic metric space, is a metric space of the form, , in which the distance between any pair of points is achieved by a minimizing geodesic. See [BBI01] for more details about length structures.
Given a pair of geodesic spaces, with base points for , one may produce a new geodesic space by identifying with . We denote this by
Consider the example of the sphere with the restricted Euclidean metric . By the Law of Cosines:
This is biLipschitz equivalent to with Lipschitz constants and . However, is not a geodesic space. In fact, for any pair of distinct points , there is no geodesic connecting and . Indeed, there cannot exist midpoints, , such that
2.2. Review of Uniform Convergence of a metric space
Fix a space . A sequence of metrics , , on is said to converge uniformly to a metric on if if
as . A sequence of metric spaces converges uniformly to a metric space if there exists a sequence of metrics on such that is isometric to , for all , and converges uniformly to .
2.3. Review of the Gromov-Hausdorff Distance
A distance preserving map, is a map such that
Observe that the standard embedding is distance preserving when viewed as but is not distance preserving when is given the length metric .
The Gromov-Hausdorff distance between two metric spaces is defined as
where the infimum is taken over all metric spaces and over all distance preserving maps . Here the Hausdorff distance between subsets is defined by
Clearly, one may estimate the Hausdorff distance from above by constructing a common space with distance preserving embeddings of and . For instance, we may estimate the distance to a single point space, :
taking to be the standard embedding. However, does not have a distance preserving map into .
Given a metric space, , let be the maximum number of disjoint balls of radius in . Gromov’s Compactness Theorem states that any sequence of metric spaces, , with a uniform upper bound, , for all , has a subsequence which converges in the Gromov-Hausdorff sense to a compact metric space. He proved the converse as well: if compact converges to a compact in the Gromov-Hausdorff sense, then there is uniform upper bound, [Gro81]. As a final remark, if a metric space is the Gromov-Hausdorff limit of geodesic spaces, then is a geodesic space as well ([Gro81]).
2.4. Review of Integral Current Spaces
An -dimensional integral current space is a metric space equipped with an -dimensional integral current . For the special case where is an oriented smooth -dimensional manifold and is a Lipschitz distance function on , then has a cannonical -dimensional integral current space structure given by
for any -form on . For instance, and are both integral current spaces.
When is only a metric space, however, there is no general notion of a smooth differential form. To define an integral current structure on such an , one applies the methods of DiGeorgi [DeG95] and Ambrosio-Kirchheim [AK00] to replace the role of smooth -forms by -tuples of Lipschitz functions . The notion of an integral current acting on such tuples is developed in detail in [AK00]. So long as is covered almost everywhere by a countable collection of biLipschitz charts , with disjoint images, then
defines an integral current on .
The mass, mass measure, and density of an integral current space are defined in [SW11], based upon [AK00]. For the purposes of this paper, we will only consider cases where is an oriented -dimensional Riemannian manifold or is glued together from such manifolds. For these cases, the mass of is , the mass measure is the -dimensional Hausdorff measure , and the density is
Since integral current spaces are defined in [SW11] so that
we see that only top dimensional regions in spaces created by gluing together manifolds form part of .
2.5. The Intrinsic Flat Distance:
The intrinsic flat distance between two integral current spaces was defined by the second author and Wenger in [SW11] imitating Gromov’s definition of the Gromov-Hausdorff distance:
where the infimum of the flat distance, , is taken over all integral current spaces and all distance preserving maps . Here is the pushforward of the current structure to a current on defined by
Recall the flat distance between two currents on an integral current space , is
where is an -dimensional integral current on and is -dimensional.
To estimate the intrinsic flat distance between two -dimensional integral current spaces, one constructs a metric space which is glued together from pieces which are biLipschitz -dimensional manifolds , such that
Here consists of the leftover pieces of boundary. If this is done in an oriented way so that
then, since and have weight one, we have
For example, if we wish to estimate the intrinsic flat distance between and , then we can consider the metric space defined in (LABEL:Back7) to see that
As in the Gromov-Hausdorff setting, it may be difficult to construct integral current spaces with distance preserving maps , , that give a sharp estimate on the distance between the spaces being considered.
The intrinsic flat distance between two integral current spaces is if and only if there is a current preserving isometry between them, see [SW11]. Thus, for example,
For a sequence of integral current spaces, the Gromov-Hausdorff and Intrinsic Flat limits do not necessarilly agree. However, if a sequence of integral current spaces converges in the Gromov-Hausdorff sense then a subsequence converges in the intrinsic flat sense to a subset of the Gromov-Hausdorff limit, see [SW10]. For example, the pair of spheres joined by increasingly thin tunnels converges in the Gromov-Hausdorff sense to a pair of spheres joined by a thread, of [SW11]. The same sequence converges in the intrinsic flat sense to the pair of spheres with the thread removed .
2.6. The Gromov-Lawson construction
A key ingredient in the proof of Theorem 1.1 is the construction of particular Riemannian metrics with positive scalar curvature on the cylinder for . These metrics smoothly transition between a small round metric on and the boundary of a ball in an arbitrary Riemanian manifold of positive scalar curvature on . Gluing two such cylinders together along the round boundary components, one obtains a long and thin tunnel – called Gromov-Lawson tunnels – transitioning between two balls in positive scalar curvature manifolds. This allows one, for instance, to produce metrics of positive scalar curvature on the connected sum of manifolds with positive scalar curvature. Although the original construction is due to Gromov-Lawson [GL80], we will require a recent generalization due to Dodziuk [Dod18].
For our constructions in Sections 3 and 4, we will require a detailed description of the construction when applied to manifolds which have regions isometric to geodesic balls in the standard sphere. For such an -dimensional positive scalar curvature manifold , fix two points and choose a radius so that the geodesic balls and are disjoint and isometric to geodesic balls in the standard sphere. Topologically, one can perform surgery on the points by removing the balls and attaching the cylinder to the resulting geodesic sphere to obtain
The Gromov-Lawson construction provides a Riemannian metric on which smoothly attaches to on , producing a positive scalar curvature metric on . In [Dod18], a refinement of the construction is given in order to obtain a qualitative description of the geometry of the metric on .
[Dod18, Proposition 1] Let and be as above, and let . There is a number and a Riemannian metric on satisfying the following:
has positive scalar curvature;
and can be glued together to form a smooth metric on
if is a continuous curve connecting the two boundary components of , then the tubular neighborhood of of radius contains i.e.
Moreover, can be chosen arbitrarily small and the estimates in items and depend only on the geometry of and .
For given parameters and , there exists a number and a function so that
and the metric in Theorem 2.1 is the restriction of the flat metric on . Notice that , but the length of the graph of is .
For clarity, there are unit spheres on the -axis equipped with geodesic balls, and , so that
is a smooth hypersurface of . Notice that is rotationally symmetric in the sense that the natural action of is isometric.
3. Revised Pipe Filling Technique in Positive Scalar Curvature
In this section, we state and prove our main technical result, Theorem 3.1. In it, we clarify and expand upon the pipe filling technique originally introduced by the third named author in the appendix to [SW11]. More details are given here that could be used to clarify the construction in that appendix, as well. To write a detailed proof and to incorperate the positive scalar curvature condition, we change the technique somewhat.
Let us introduce the setting for the pipe filling technique. For , let be an oriented closed -dimensional Riemannian manifold with positive scalar curvature. Suppose we are given points and a radius so that and are isometric to geodesic balls in the unit sphere . We also fix a length . According to Theorem 2.1, there is number and a manifold which is topologically , has positive scalar curvature, and smoothly glues to to form
Denote the corrisponding integral current space by .
Following Remark 2.2, there is a function which, upon properly rotating its graph, produces the Gromov-Lawson tunnel . We emphasize that the function depends only on and is independant of the ambient geometry of .
We also let denote the space obtained by joining and by a line segment
Denote the corrisponding integral current space by .
Let an oriented closed -dimensional Riemannian manifold with positive scalar curvature. Let and be so that and are isometric to a spherical geodesic ball . Fix and let and be the resulting integral current spaces described above in (3.1) and (3.2).
Then there is a constant , continuously depending only on , , and , so that
The remainder of this section is devoted to the proof of Theorem 3.1.
3.1. Constructing the common space
We begin by constructing an integral current space with boundary which we will use to estimate the flat and Gromov-Hausdorff distances. To begin, consider the product
where and are given by
The next step is to produce a space which will glue into along its boundary.
Following Remark 2.2, the Gromov Lawson tunnel can be viewed as a hypersurface in which transitions between two unit spheres equipped with spherical geodesic balls
The space , pictured in Figure 3, will be constructed in three portions
The first piece is a product of the Gromov-Lawson tunnel with the segment ,
The second piece consists of two parts: half of a rotated Gromov-Lawson tunnel we call the pipe, , and a cuspoidal region, as in Figure 3. More precisely,
Notice that the union is a -hypersurface of .
The third and final piece, , is given by
where the line segment is attached to the centers of and . Notice that the subset
may be isometrically identified with the top portion of so that the union is a well-defined integral current space.
Finally, a portion of the boundary of can be isometrically identified with . Gluing along this edge, we form the integral current space .
Notice that there is a (non-continuous) height function and that the hypersurfaces and can be identified with and , repsectively. Let and denote the corresponding inclusions. We also denote . Using the volume estimate of Theorem 2.1 and Remark 2.2, there is a constant , independant of and , so that
3.2. The embeddings and are distance preserving
The embedding is distance preserving.
Assume on the contrary that there exists points such that
Then there exists an arclength parametrized curve such that
For ease of notation, we will use to denote both the parameterization and its image. The remainder of the proof will be broken into claims.
intersects the set
Inspecting the definition of , there is a distance nonincreasing map from to and so we may assume that lies in .
Let be the continuous distance nonincreasing map which takes
to . If avoids the pipe, then is a curve running between and whose length is no greater than and lies entirely in , which is a contradiction. It follows that .
Now let be the continuous distance nonincreasing map defined by sending
to a point in the image of the thread:
Let be defined so that for and for . This map is continuous everywhere except on the set . If our curve avoids this set, then is a shorter curve between its endpoints whose image lies in which is a contradiction. This establishes Claim 1. ∎
Let , , and where is the component of closest to . Then we have
Our next task is to produce a competitor to . According to Claim 1 we may consider the first and final times – denoted by and , respectively – the path passes through the set . We replace the segment by a minimizing -geodesic running directly from to . Likewise, we replace the segment by a minimizing -geodesic running directly from to . By Claim 2, we know that and .
Now we have two possibilities: either or . If , then we may form the concatination which is path from to lying entirely within and has length . This contradicts our assumption and we are done. Now assume that . We can consider a minimizing -geodesic which runs from to . Notice that , which can be seen using . It follows that the concatination is a path from to lying entirely within and has length , also yielding the desired contradiction. ∎
The map is distance preserving.
Assume on the contrary that there exists such that
Then there exists an arclength parametrized curve such that and and . As before, we will break the remainder of the proof into smaller claims.
We may assume that and must intersect the set .
We first notice that we may assume . This can be seen by applying the distance nonincreasing map defined by to any portion of lying above height . On the other hand, there is a projection mapping which is distance nonincreasing due to the product structure of there. It follows that must pass through the set . Since begins and ends at height , must intersect . ∎
The curve can be replaced by a curve satisfying the following
does not pass through the interior of the region ;
If passes through , we replace this segment of by a path lying entirely within . Since the diameter of is less than , this replacement yields a curve