Papers on the Intrinsic Flat Distance between Riemannian Manifolds

ordered according to availability as a preprint on the arxiv 

with abstracts and

with comments as to relevance and correctness added over time

2009 Geometry Festival Presentation and other presentations

please let me know if there are missing papers

sormanic at gmail 

[SW11] "The Intrinsic Flat Distance between Riemannian Manifolds and Integral Current Spaces"

by Christina Sormani and Stefan Wenger (arxiv preprint) 

Journal of Differential Geometry, Vol 87, (2011) (reprint)

Abstract: Inspired by the Gromov-Hausdorff distance, we define the intrinsic flat distance between oriented m dimensional Riemannian manifolds with boundary as:

                                  dF(M,N) = inf dZF ( [f(M)] , [h(N)] )

where f,h are isometric embeddings from M and N respectively into a complete metric space Z and where [f(M)] and [h(N)] denote their images viewed as integral currents in Z in the sense of Ambrosio-Kirchheim and the infimum is taken over all $f$, $h$ and $Z$.

We show the intrinsic flat distance between oriented Riemannian manifolds is zero iff they have an orientation preserving isometry between them. Using the theory of Ambrosio-Kirchheim, we study converging sequences of manifolds and their limits, which are in a class of metric spaces that we call integral current spaces. We describe the properties of such spaces including the fact that they are countably Hm rectifiable spaces and give numerous examples.

Comments on prior work leading to this paper:

Whitney first introduced the flat and sharp norms for submanifolds of Euclidean space viewed as chains in his 1957 book Geometric Integration Theory.    In Federer-Flemming’s “Normal and Integral Currents” Annals of Math 2 72 (1960) 458-520 the notion of a submanifold in Euclidean space as a current acting on differential forms was introduced.  They proved  flat and weak compactness theorems and slicing theorems.   

In Ambrosio-Kirchheim’s “Currents in Metric Spaces” Acta Math Vol 185 2000 no 1, 1--80 (reprint), integral currents on a complete metric space were introduced using DeGiorgi tuples. They proved compactness theorems for sequences of currents with respect to weak convergence and much more.  In  “Flat Convergence for Integral Currents” Wenger studied flat convergence of Ambrosio-Kirchheim’s integral currents in a fixed metric space and proved when flat and weak convergence agree.

The Intrinsic Flat distance is a distance between pairs of distinct Riemannian manifolds and, more generally, integral current spaces.  It is defined by imitating Gromov’s definition of his “Intrinsic Hausdorff distance” (now called the Gromov-Hausdorff distance) using Gromov’s idea of taking an infimum over all isometric embeddings into all possible common metric spaces and then taking the Hausdorff distance between the images.  See Gromov’s 1981 book Structures Metriques pour les Varietes Riemannians.   Instead of taking the Hausdorff distance, Sormani and Wenger take the flat distance between the images.

[W11] "Compactness for manifolds and integral currents with bounded diameter and volume"

by Stefan Wenger  (arxiv preprint)

Calculus of Variations and Partial Differential Equations, 40 (2011). 

Abstract: By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class or oriented $k$-dimensional Riemannian manifolds and, more generally, integral currents in metric spaces (in the sense of Ambrosio-Kirchheim) and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which we replace equicompactness with uniform upper bounds on volume and diameter for the sequence.  Comments: In particular, Wenger has shown that any sequence of oriented $k$ dimensional Riemannian manifolds with a uniform upper bound on volume and on diameter has a subsequence which converges in the intrinsic flat sense to an integral current space. If the manifolds have boundary, he only needs to add an assumption that the boundaries have a uniform upper bound on volume.

[SW10] "Cancellation under weak convergence" (prior name: "Cancellation under flat convergence")

by Christina Sormani and Stefan Wenger (arxiv preprint) 

Appendix by Raanan Schul and Stefan Wenger

Calculus of Variations and Partial Differential Equations, Vol 38, No. 1-2, May 2010. (reprint)

Abstract: This paper concerns cancellation and collapse when a sequence of manifolds or integral currents is converging in the flat norm. Applying Gromov's filling paper and imitating a theorem of Greene-Petersen, we show that the flat limits and Gromov-Hausdorff limits of linearly locally contractible manifolds agree. As a consequence the limits of these spaces are countably Hm rectifiable spaces. Applying Cheeger-Colding and Perelman, we show that the flat limits and Gromov-Hausdorff limits of noncollapsing sequences of manifolds with nonnegative Ricci curvature agree. Cheeger-Colding had already shown that limits of such sequences are as rectifiable as current spaces. We give examples of sequences with positive scalar curvature where they do not agree. These examples have lots of local topology. Within our proofs we also describe the sets of limits of flat converging sequences of integral currents using the theory of Ambrosio-Kirchheim.  Typo: In statement of Theorem 4.1 it must explicitly require spt of the boundary to avoid the ball. This is the case in all our applications within the paper.   

[LW11] "The pointed flat compactness theorem for locally integral currents"

by Urs Lang and Stefan Wenger (arxiv preprint)

Communications in Analysis and Geometry. 19 (2011), no. 1, 159-189. 

Abstract: Recently, an embedding/compactness theorem for integral currents in a sequence of metric spaces has been established by the second author. We present a version of this result for locally integral currents in a sequence of pointed metric spaces. To this end we introduce another variant of the Ambrosio-Kirchheim theory of currents in metric spaces, including currents with finite mass in bounded sets.  Comments: this is not an intrinsic paper, as the balls employed to define the pointed convergence sit in the extrinsic space $Z$ into which the spaces are isometrically embedded. It is built on Lang's local theory of currents (rather than Ambrosio - Kirchheim's theory) to handle currents of infinite mass.   An intrinsic way of defining pointed intrinsic flat convergence of Riemannian manifolds imitating Gromov's pointed convergence, is to restrict to balls in the manifolds and study their intrinsic flat limits as balls in the limit space. 

[LS14] "Stability of the Positive Mass Theorem for Rotationally Symmetric Riemannian Manifolds"

by Dan Lee and Christina Sormani (arxiv preprint)

Journal fur die Riene und Angewandte Mathematik (Crelle's Journal), Vol 686 (January 2014)  

Abstract: We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal hypersurfaces or are empty. We prove that a sequence of these manifolds whose ADM masses converge to zero must converge to Euclidean space in the pointed Intrinsic Flat sense. In fact we provide explicit bounds on the Intrinsic Flat Distance between annular regions in the manifold and annular regions in Euclidean space by constructing an explicit filling manifold and estimating its volume. In addition, we include a variety of propositions that can be used to estimate the Intrinsic Flat distance between Riemannian manifolds without rotationally symmetry. Conjectures regarding the Intrinsic Flat stability of the Positive Mass Theorem in the general case are proposed in the final section.    

[LS12] "Almost Equality in the Penrose Equality for Rotationally Symmetric Riemannian Manifolds"

by Dan Lee and Christina Sormani (arxiv preprint)

Annales Henri Poincare November 2012, Volume 13, Issue 7, pp 1537-1556. 

Abstract: This article is the sequel to our previous paper [LS] dealing with the near-equality case of the Positive Mass Theorem. We study the near-equality case of the Penrose Inequality for the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature whose boundaries are outermost minimal hypersurfaces. Specifically, we prove that if the Penrose Inequality is sufficiently close to being an equality on one of these manifolds, then it must be close to a Schwarzschild space with an appended cylinder, in the sense of Lipschitz Distance. Since the Lipschitz Distance bounds the Intrinsic Flat Distance on compact sets, we also obtain a result for Intrinsic Flat Distance, which is a more appropriate distance for more general near-equality results.  

[LS13] "Smooth Convergence Away from Singular Sets"

by Sajjad Lakzian and Christina Sormani

Communications in Analysis and Geometry, Volume 21 (2013) No 1, 39-104 (arxiv v4 is better than the publication)

Abstract: This paper applies intrinsic flat convergence to understand smooth convergence away from singular sets of codimension two and to determine when the metric completion of a smooth limit agrees with the Gromov-Hausdorff or Intrinsic Flat limit of the manifolds. Many examples are given.   Comments: Brian Allen found a counter example to the original statement Theorem 1.3 after the paper was published in CAG.   Note that Theorem 4.6, the main theorem applied in papers citing this one, was correct as originally stated and proven.  The paper has been reposted on the arxiv with an additional hypotheses in Theorem 1.3 and its consequences.  The v4 on the arxiv also has an appendix with the counter example.  An erratum has been published in the journal.

[G12] "Hilbert Volume in Metric Spaces, Part I"

by Misha Gromov (reprint) (arxiv post later)

Central European Journal of Mathematics April 2012, Volume 10, Issue 2, pp 371–400 

Abstract: We introduce a notion of Hilbertian n-volume in metric spaces with Besicovitch-type inequalities built-in into the definitions. The present Part 1 of the article is, for the most part, dedicated to the reformulation of known results in our terms with proofs being reduced to (almost) pure tautologies. If there is any novelty in the paper, this is in forging certain terminology which, ultimately, may turn useful in an Alexandrov kind of approach to singular spaces with positive scalar curvature [Gromov M., Plateau-hedra, Scalar Curvature and Dirac Billiards, in preparation].   Comments: The paper states that the (1+epsilon) biLipschitz construction is too strong and upcoming work will consider the intrinsic flat distance.

[L16] "Diameter Controls and Smooth Convergence away from Singular Sets"

by Sajjad Lakzian (arxiv preprint) (reprint)

Journal of Differential Geometry and Applications, Volume 47, August 2016, Pages 99-129 

Abstract: This paper builds upon and extends prior results of the author and C. Sormani considering both the Intrinsic Flat and Gromov-Hausdorff limits of sequences of manifolds with metrics that converge smoothly away from singular sets $S\subset M^m$ of Hausdorff measure $H^{m-1}(S)=0$.  Comments: The error in [LS13] may propagate to this paper.  An erratum is in progress which will add additional hypotheses.

[L15] ``Continuity of Ricci flow through Neck Pinch Singularities"

by Sajjad Lakzian  (arxiv preprint) (reprint)

Geometriae Dedicata, December 2015, Volume 179, Issue 1, pp 69–89 

Abstract: In this article, we consider the Angenent-Caputo-Knopf's Ricci Flow through neckpinch singularities. We will explain how one can see the A-C-K's Ricci flow through a neckpinch singularity as a flow of integral current spaces. We then prove the continuity of this weak flow with respect to the Sormani-Wenger Intrinsic Flat (SWIF) distance.

[G14] ``Dirac and Plateau Billiards in Domains with Corners"

by Misha Gromov (posted preprint, (Sept 30, 2013))  (arxiv preprint later)

Central European Journal of Mathematics 12 (2014): 1109-1156.

Abstract: Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C^2 -smooth Riemannian metrics g on a smooth manifold X, with scalar curvature bounded from below by a continuous function is closed under C^0 limits of Riemannian metrics for all continuous functions on X. Apart from that our progress is limited but we formulate many conjectures. All along we emphasize geometry, rather the the topology of the manifolds with their scalar curvature bounded from below.  Comments: Within this paper, Gromov suggests that a sequence of tori with Riemannian metrics g_k whose scalar curvature is bounded below by -1/k might be shown to converge in the intrinsic flat sense to a flat torus as k diverges to infinity. The geometric properties of scalar curvature that Gromov studies in this paper may be useful towards proving such a theorem.

[P15] "Semicontinuity of eigenvalues under intrinsic flat convergence"

by Jacobus W. Portegies (arxiv preprint)

Calculus of Variations and Partial Differential Equations October 2015, Volume 54, Issue 2, pp 1725-1766

Abstract: We use the theory of rectifiable metric spaces to define a normalized Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. We define min-max values based on the normalized energy and show that when integral current spaces converge in the intrinsic flat sense without loss of volume, the min-max values of the limit space are larger than or equal to the upper limit of the min-max values of the currents in the sequence. In particular, the infimum of the normalized energy is semicontinuous. On spaces that are infinitesimally Hilbertian, we can define a linear Laplace operator. We can show that semicontinuity under intrinsic flat convergence holds for eigenvalues below the essential spectrum, if the total volume of the spaces converges as well.    The appendix contains a decomposition theorem for one dimensional integral currents and a Poincare-like inequality.

[G14] "Plateau Stein Manifolds"

by Misha Gromov, ihes preprint (arxiv preprint later)

Central European Journal of Mathematics July 2014, Vol 12, Issue 7, pp 923-951

Abstract: We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds $X$ such that the hypersurfaces f(x) = t for all −∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f. We show, for instance, that if an X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.  Comments: In this paper Gromov suggests the intrinsic flat distance might be useful to study sequences of manifolds with nonnegative scalar curvature.

[S19] "Intrinsic Flat Arzela-Ascoli Theorems"

by Christina Sormani, arxiv preprint, 

Communications in Analysis and Geometry Vol. 27, No 1, 2019 (reprint)

Abstract: In this paper two Arzela-Ascoli Theorems are proven: one for uniformly Lipschitz functions whose domains are converging in the intrinsic flat sense, and one for sequences of uniformly local isometries between spaces which are converging in the intrinsic flat sense. A basic Bolzano-Weierstrass Theorem is proven for sequences of points in such sequences of spaces. In addition it is proven that when a sequence of manifolds has a precompact intrinsic flat limit then the metric completion of the limit is the Gromov-Hausdorff limit of regions within those manifolds. Open problems with suggested applications are provided throughout the paper.

[LS15] "Nonlinear stability of rotationally symmetric spaces with low regularity"

by Philippe LeFloch and Christina Sormani, (arxiv preprint) 

Journal of Functional Analysis  Vol. 268 (2015), no. 7, 2005–2065.  

Abstract: Our main theorem states that if one has a sequence of rotationally symmetric regions of nonnegative scalar curvature with no closed interior minimal surfaces that have spherical boundaries of fixed area with uniform upper bounds on:

• • • the depth of the region and

    • the Hawking mass of the boundary,

then a subsequence converges in the intrinsic flat sense to a region whose metric h\ as $H^1$ regularity with nonnegative scalar curvature (in the generalized sense). \ In addition

• • • the area of the boundary is continuous,

    • the volume of the region is continuous and

    • the Hawking mass of the boundary is continuous.

In order to obtain the continuity of these values we show the metric tensors converge in the Sobolev sense.

In addition to this main theorem, we provide a thorough analysis of these limit spaces, proving the Hawking mass is monotone and providing precise estimates on the intrinsic flat distance, the Sobolev distance, and a new notion, the D flat distance, between a region of small Hawking mass and a disk in Euclidean space extending results of Sormani with Dan Lee. It should be noted that the ideas involving the generalized notions of curvature on a manifold with $H^1$ metric tensors has been explored by LeFloch and collaborators Mardare, Stewart and Rendall. Ordinarily intrinsic flat limit spaces are only countably H^m rectifiable.   

[J18] "On Lower Semicontinuity of the ADM mass"

by Jeffrey L. Jauregui,  (arxiv preprint) 

Communications in Analysis and Geometry Vol 26 issue 1 (2018) 18 pages 

Abstract: The ADM mass, viewed as a functional on the space of asymptotically flat Riemannian metrics of nonnegative scalar curvature, fails to be continuous for many natural topologies. In this paper we prove that lower semicontinuity holds in natural settings: first, for pointed Cheeger--Gromov convergence (without any symmetry assumptions) for n=3 and , second, assuming rotational symmetry, for weak convergence of the associated canonical embeddings into Euclidean space, and, finally, also assuming rotational symmetry, for pointed intrinsic flat convergence (applying work of LeFloch-Sormani).  We provide several examples, one of which demonstrates that the positive mass theorem is implied by a statement of the lower semicontinuity of the ADM mass.    

[P16] "Volumes and Limits of Manifolds with Ricci Curvature and Mean Curvature Bounds"

by Raquel Perales, (arxiv preprint) 

Journal of Differential Geometry and Applications  Vol 48, (2016) pp 23-37 (reprint)

Abstract: This paper proves a number of theorems including the following intrinsic flat compactness theorems building upon Wenger's Compactness Theorem: A sequence of n dimensional oriented manifolds with boundary that have nonnegative Ricci curvature and an upper bound on mean curvature and area of the boundary and an upper bound on diameter, have a subsequence which converges in the intrinsic flat sense. The diameter bound on the manifold may be replaced by a diameter bound on the boundary if the mean curvature is uniformly strictly negative in this theorem. The necessity of various conditions have been presented as examples in this paper as well. It is unknown if such a sequence has a subsequence converging in the GH sense or whether GH and intrinsic flat limits agree in this setting.

[M16] "Intrinsic Flat Convergence with bounded Ricci curvature"

by Michael Munn, arxiv preprint (never published: Munn left academia)

Abstract: In this paper we address the relationship between Gromov-Hausdorff limits and intrinsic flat limits of complete Riemannian manifolds. In prior work, Sormani-Wenger show that for a sequence of Riemannian manifolds with nonnegative Ricci curvature, a uniform upper bound on diameter, and non-collapsed volume, the intrinsic flat limit exists and agrees with the Gromov-Hausdorff limit. This can be viewed as a non-cancellation theorem showing that for such sequences, points don't cancel each other out in the limit. Here we prove a similar no-cancellation theorem, replacing the assumption of nonnegative Ricci curvature with a two-sided bound on Ricci curvature.  Comments: Never published when Munn left academia for industry. Stronger results have since been prove in work of Matveev-Portegies linked below.

[HLS15]  "Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space"

by Lan-Hsuan Huang, Dan Lee, Christina Sormani, arxiv preprint 

Crelle Journal für die reine und angewandte Mathematik  727 (2015)  (arxiv preprint)  (reprint)

Abstract: The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this statement for spaces that can be realized as graphical hypersurfaces in Euclidean space. We prove (under certain technical hypotheses) that if a sequence of complete asymptotically flat graphs of nonnegative scalar curvature has mass approaching zero, then the sequence must converge to Euclidean space in the pointed intrinsic flat sense. The appendix includes a new Gromov-Hausdorff and intrinsic flat compactness theorem for sequences of metric spaces with uniform Lipschitz bounds on their metrics.  Comments: In 2020 Cabrera-Pacheco and Perales questioned a step in the proof near the very end of the paper.  Huang, Lee, and Perales wrote a paper providing a new proof of the results using techniques developed by Allen-Perales here in the place of Wenger's Compactness Theorem.  See (preprint).   In 2022 Del Nin and Perales (arXiv:2210.06406) have proven the original proof holds using new theory on integral current spaces to adequately justify the steps.

[SS16] "Intrinsic Flat Convergence of Covering Spaces"

by Zahra Sinaei and Christina Sormani,  (arxiv preprint)

Geometriae Dedicata, Vol 184 (2016) pages 83–114  (reprint) 

Abstract: We examine the limits of covering spaces and the covering spectra of oriented Riemannian manifolds, Mj, which converge to a nonzero integral current space, M∞, in the intrinsic flat sense. We provide examples demonstrating that the covering spaces and covering spectra need not converge in this setting. In fact we provide a sequence of simply connected Mj diffeomorphic to S4 that converge in the intrinsic flat sense to a torus S1×S3. Nevertheless, we prove that if the delta covers have finite order N, then a subsequence of these delta covers converge in the intrinsic flat sense to a metric space which is the disjoint union of covering spaces of M∞.   This is related to prior work of the the second author with Guofang Wei.

[LP 18] "On the Sormani-Wenger Intrinsic Flat Convergence of Alexandrov Spaces"

by Nan Li and Raquel Perales (arxiv preprint) 

Journal of Topology and Analysis (2018)

Abstract: We study noncollapsing sequences of integral current spaces with no boundary that have lower Alexandrov curvature bounds and weight 1.   We prove that for such sequences Sormani-Wenger Intrinsic Flat convergence and Gromov-Hausdorff convergence agree.

[PS17] "Properties of the Intrinsic Flat Distance"

by Jacobus Portegies and Christina Sormani, arxiv preprint, solicited for the Volume in Honor of Yuri Burago, editted by D Burago and S Buyalo,  Algebra i Analiz Issue 3, Volume 29 (2017), pages 70-143). (DMS 1006059)

Abstract: Here we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the disappearance of points under intrinsic flat convergence. We prove two new Gromov-Hausdorff and intrinsic flat compactness theorems including the Tetrahedral Compactness Theorem. Much of the work in this paper builds upon Ambrosio-Kirchheim's Slicing Theorem combined with an adapted version Gromov's Filling Volume.

[P18] "Convergence of Manifolds and Metric Spaces with Boundary"

by Raquel Perales,  (arxiv preprint) (completed in May 2015) 

Journal of Topology and Analysis 12.03 (2020): 735-774.

Abstract: We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. We prove theorems demonstrating when the Gromov-Hausdorff [GH] and Sormani-Wenger Intrinsic Flat [SWIF] limits of sequences of such metric spaces agree. Thus in particular the limit spaces are countably H^n rectifiable spaces. From these theorems we derive compactness theorems for sequences of Riemannian manifolds with boundary where both the GH and SWIF limits agree. For sequences of Riemannian manifolds with boundary we only require nonnegative Ricci curvature, upper bounds on volume, noncollapsing conditions on the interior of the manifold and diameter controls on the level sets near the boundary.   Note: This paper also contains a nice review of the Sormani-Wenger theorem for manifolds with nonnegative Ricci curvature that have no boundary.

[MP17] "Intrinsic Flat and Gromov-Hausdorff Convergence for Manifolds with Ricci curvature bounded below"

by Rostitslav Matveev and Jacobus Portegies, (arxiv),(completed in Oct 2015) (reprint)

The Journal of Geometric Analysis Vol 27 issue 3 (July 2017) 1855-1873.

Abstract: We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with intrinsic flat convergence.   Comments: This paper provides a new proof for the Ricci nonnegative case as well.

[S17] "Scalar Curvature and Intrinsic Flat Convergence"

by Christina Sormani, (preprint) These are the notes for the Como School. (reprint)

published as a chapter in Measure Theory in Non-Smooth Spaces edited by Nicola Gigli de Gruyter Press (2017)

Abstract:  Herein we present open problems and survey examples and theorems concerning sequences of Riemannian manifolds with uniform lower bounds on scalar curvature and their limit spaces. Examples of Gromov and of Ilmanen which naturally ought to have certain limit spaces do not converge with respect to smooth or Gromov-Hausdorff convergence. Thus we focus here on the notion of Intrinsic Flat convergence, developed jointly with Wenger. This notion has been applied successfully to study sequences that arise in General Relativity. Gromov has suggested it should be applied in other settings as well. We first review intrinsic flat convergence, its properties, and its compactness theorems, before presenting the applications and the open problems.  Comments: In this paper the term "almost rigidity" is used in a more general way than it was used in the original 1996 Cheeger-Colding paper coining the term.  In April 2018, Cheeger stated that he was not happy with this and  requested that the term not be used in this way in future work.  Additional Comments: This paper includes a description of papers listed above some of which now have errata.  

[H17] "Ricci Curvature and Orientability"

by Shouhei Honda, (arxiv preprint), CVPDE 56 (2017) (reprint)

Abstract:  In this paper we define an orientation of a measured Gromov-Hausdorff limit space of Riemannian manifolds with uniform Ricci bounds from below. This is the first observation of orientability for metric measure spaces. Our orientability has two fundamental properties. One of them is the stability with respect to noncollapsed sequences. As a corollary we see that if the cross section of a tangent cone of a noncollapsed limit space of orientable Riemannian manifolds is smooth, then it is also orientable in the ordinary sense, which can be regarded as a new obstruction for a given manifold to be the cross section of a tangent cone. The other one is that there are only two choices for orientations on a limit space. We also discuss relationships between L2-convergence of orientations and convergence of currents in metric spaces. In particular for a noncollapsed sequence, we prove a compatibility between the intrinsic flat convergence by Sormani-Wenger, the pointed flat convergence by Lang-Wenger, and the Gromov-Hausdorff convergence, which is a generalization of a recent work by Matveev-Portegies to the noncompact case. Moreover combining this compatibility with the second property of our orientation gives an explicit formula for the limit integral current by using an orientation on a limit space. Finally dualities between de Rham cohomologies on an oriented limit space are proven.  

[BDS18] "Sewing Riemannian Manifolds with Positive Scalar Curvature"

by Jorge Basilio, Jozef Dodziuk and Christina Sormani,   (arxiv) 

Journal of Geometric Analysis 28 (2018)  (reprint) (DMS 1006059)

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.   

[JPRSS] "Alexandrov Spaces with Integral Current Structure"

by Maree Jaramillo, Raquel Perales, Priyanka Rajan, Catherine Searle, Anna Siffert  (arxiv) 

Communications in Analysis and Geometry 29.1 (2021).

Abstract: We endow each closed, orientable Alexandrov space (X,d) with an integral current T of weight equal to 1, $\partial T = 0 and \set(T) = X$, in other words, we prove that (X,d,T) is an integral current space with no boundary. Combining this result with a result of Li and Perales (arxiv preprint), we show that non-collapsing sequences of these spaces with uniform lower curvature and diameter bounds admit subsequences whose Gromov-Hausdorff and intrinsic flat limits agree.   This builds upon a paper of Mitsuishi (arxiv preprint) which does not involve integral current spaces but does apply Ambrosio-Kirchheim theory to Alexandrov spaces. (NSF DMS 1309360) (team website)

[SS17] "Almost Rigidity of the Positive Mass Theorem for Asymptotically Hyperbolic Manifolds with Spherical Symmetry"

by Anna Sakovich and Christina Sormani, (arxiv preprint) 23 pages, 

General Relativity and Gravitation, selected to be an Editor's Choice, September 2017 49:125. (reprint)

We use the notion of intrinsic flat distance to address the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. In particular, we prove that a sequence of spherically symmetric asymptotically hyperbolic manifolds satisfying the conditions of the positive mass theorem converges to hyperbolic space in the intrinsic flat sense, if the limit of the mass along the sequence is zero.  (NSF DMS 1612409)  Comments: In this paper the term "almost rigidity" is used in a more general way than it was used in the original 1996 Cheeger-Colding paper coining the term.  In April 2018, Jeff Cheeger stated he was not happy with this and requested that the term not be used in this way in future work.

[LM] "Positive Scalar curvature with Skeleton Singularities"

by Chao Li and Christos Mantoulodis, (arxiv preprint) 

Mathematische Annalen, Vol 374 (2019) (reprint)

We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean metrics that consolidate Gromov's scalar curvature polyhedral comparison theory and edge metrics that appear in the study of Einstein manifolds. We show that, in all dimensions, edge singularities with cone angles le 2π along codimension-2 submanifolds do not affect the Yamabe type. In three dimensions, we prove the same for more general singular sets, which are allowed to stratify along 1-skeletons, exhibiting edge singularities (angles le 2π) and arbitrary L∞ isolated point singularities. We derive, as an application of our techniques, Positive Mass Theorems for asymptotically flat manifolds with analogous singularities.  Comments: This has a conjecture concerning intrinsic flat convergence and their resolution of singularities in Section 8.

[SS19] "Geometrostatic manifolds of small ADM Mass"

by C Sormani and I Stavrov, 42 pages, (arxiv preprint) 

Communications in Pure and Applied Mathematics Vol 72 (2019) (reprint)

We bound the locations of minimal surfaces in Geometrostatic manifolds of small ADM mass and then prove the Intrinsic Flat Stability of the Positive Mass Theorem in the Geometrostatic setting. (NSF DMS 1309360)  

[NP] "A generalized tetrahedral property for spaces with conical singularities"

by J Nunez-Zimbron and R Perales (arxiv preprint) 

Mathematische Zeitschrift October (2020)

We extend the definition of Sormani's Tetrahedral Property so that conical metric spaces satisfy our new definition. We prove that our generalized definition retains all the properties of the original tetrahedral property proven by Portegies-Sormani: it provides a lower bound on the sliced filling volume and Gromov filling volume of spheres, and a lower bound on the volumes of balls. Thus, sequences with uniform bounds on our generalized tetrahedral property also have subsequences which converge in both the Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat sense to the same limit space. (NSF DMS 1309360)

[AS19] "Contrasting Various Notions of Convergence in Geometric Analysis"

by Brian Allen and Christina Sormani,  43 pages, (arxiv preprint)  

Pacific Journal of Mathematics Vol 303 (2019) (reprint)

We explore the distinctions between $L^p$ convergence of metric tensors on a fixed Riemannian manifold versus Gromov-Hausdorff, uniform, and intrinsic flat convergence of the corresponding sequence of metric spaces.   We provide a number of examples which demonstrate these notions of convergence do not agree even for two dimensional warped product manifolds with warping functions converging in the $L^p$ sense.  We then prove a theorem which requires $L^p$ bounds from above and $C^0$ bounds from below on the warping functions to obtain enough control for all these limits to agree. (NSF DMS 1612049)

[AHPPW19] “Warped Tori of Almost Nonnegative Scalar Curvature" originally named "Almost Rigidity of Warped Tori"

by Brian Allen, Lisandra Hernandez-Vazquez, Davide Parise, Alec Payne, Shengwen Wang (arxiv)

Geometriae Dedicata, June 2019, Volume 200, Issue 1, p. 153-171

We show that for warped products on a 3-torus, there is almost rigidity of the Scalar Torus Rigidity Theorem: for sequences of warped product metrics on a 3-torus satisfying the scalar curvature bound Rj≥−1j, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrinsic Flat sense to a flat 3-torus.  (NSF DMS 1612049) (Team website).  Comments: Cheeger was not happy with the use of the term “almost rigidity” both in title and within the paper and requested the change. The term was first used in his 1996 paper with Colding and had a more restricted meaning there.  Although the team offered to clarify the distinction between “Cheeger Colding Almost Rigidity” and the more general modern meaning, he did not agree to this compromise.  He requested that everyone stop using “Almost Rigidity” in any setting other than his original meaning.  In deference to his seniority we should stop.

[S18]  “Oberwolfach Report: 2018 Spacetime Intrinsic Flat Convergence”

by C Sormani (arxiv)

This reports on current work with Sakovich and Vega developing a notion of spacetime intrinsic flat convergence.

 [CL]  ``Positive Scalar Curvature with Skeleton Singularities"

by Chao Li and Christos Mantoulidis,  (arxiv) 

Math Annalen 374, 99-131 (2019). (reprint)

We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean (L∞) metrics that consolidate Gromov’s scalar curvature polyhedral comparison the- ory and edge metrics that appear in the study of Einstein manifolds. We show that, in all dimensions, edge singularities with cone angles ≤ 2π along codimension-2 submanifolds do not affect the Yamabe type. In three dimensions, we prove the same for more general singular sets, which are allowed to stratify along 1-skeletons, exhibiting edge singularities (angles ≤ 2π) and arbitrary L∞ isolated point singularities. We derive, as an application of our techniques, Positive Mass Theorems for asymptotically flat manifolds with analogous singularities.   Comments: Chao Li believes that the construction in this paper shows that if a smooth manifold is equipped with a singular PSC metric with isolated singularities at which the metric is L^\infty, then it can be approximated by smooth PSC metrics, in the pointed intrinsic flat sense.   The SWIF convergence has not yet been proven explicitly but intuitively this looks right.

[A18]  “Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem”

        by Brian Allen  (arxiv)

We study the stability of the Positive Mass Theorem (PMT) in the case where a sequence of regions of manifolds with positive scalar curvature $U_T^i\subset M_i^3$ are foliated by a smooth solution to Inverse Mean Curvature Flow (IMCF) which may not be uniformly controlled near the boundary. Then if $\partial U_T^i = \Sigma_0^i \cup \Sigma_T^i$, $m_H(\Sigma_T^i) \rightarrow 0$ and extra technical conditions are satisfied we show that $U_T^i$ converges to a flat annulus with respect to Sormani-Wenger Intrinsic Flat (SWIF) convergence. 

``The pointed intrinsic flat distance between locally integral current spaces"

by Shu Takeuchi (https://arxiv.org/abs/1809.07641)

In this note we define a distance between two pointed locally integral current spaces. We prove that a sequence of pointed locally integral current spaces converges with respect to this distance if and only if it converges in the sense of Lang-Wenger. This enables us to state the compactness theorem by Lang-Wenger for pointed locally integral current spaces in terms of a distance function. 

“Sobolev bounds and convergence of Riemannian manifolds”

by Brian Allen and Edward Bryden with Appendix by Allen and Sormani (arxiv) 

Nonlinear Analysis Volume 185, August 2019, Pages 142-169

We prove that Sobolev bounds on sequences of metric tensors imply Holder bounds on their distance functions. In the Appendix we prove this Holder bounds imply a subsequence converges in the uniform, GH, and SWIF sense. We prove the metric completion of the SWIF limit is the GH limit.  (NSF DMS 1612049)

``An intrinsic flat limit of Riemannian manifolds with no geodesics''

by Jorge Basilio, Demetre Kazaras, and Christina Sormani. (arxiv)

Geometriae Dedicata Vol 204, 265-284 (2020) (reprint)

In this paper we produce a sequence of Riemannian manifolds $M_j^m$, $m \ge 2$, which converge in the intrinsic flat sense to the unit $m$-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 $m\geq3$, the manifolds $M_j^m$ may be chosen to have positive scalar curvature.  (DMS1612049 and Basilio DMS1006059)

``IAS Report on Emerging Topics in Scalar Curvature and Convergence"

by M Gromov and C Sormani  (link to IAS posting)

This is a report on the IAS Emerging Topics on Scalar Curvature and Convergence organized by Gromov and Sormani, including a conjecture framed during the conference.   More conjectures will appear in a future paper by those attending the workshop.

``Scalar Curvature of Manifolds with Boundaries: Natural Questions and Artificial Constructions''

by Misha Gromov (arxiv) (posted in various places but not published)

We present several problems and results relating the scalar curvatures of manifolds with mean curvatures of their boundaries.  I describe here in writing what came out of a conversation between Chao Li, Pengzi Miao, André Neves, Christina Sormani, and myself which took place during the workshop  Emerging Topics on Convergence and Scalar Curvature in October 15-19, 2018 at IAS in Princeton. Comments: This contains a number of conjectures and thoughts, and near the end a new conjecture Gromov calls the Spherical Stability Problem concerning intrinsic flat convergence.

“A Compactness Theorem for Rotationally Symmetric Riemannian Manifolds with Positive Scalar Curvature”

by Jiewon Park, Wenchuan Tian, Changliang Wang  (arxiv) (reprint) 

Pure and Applied Mathematics Quarterly Vol 14 (2018) no 4  pp 529-561.

Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces which have nonnegative generalized scalar curvature and Euclidean tangent cones almost everywhere. In this paper we prove this conjecture for sequences of rotationally symmetric warped product manifolds. We show that the limit spaces have $H^1$ warping function that has nonnegative scalar curvature in a weak sense, and have Euclidean tangent cones almost everywhere. 

“Initial Data and Black Holes for Matter Models"

by Annegret Burtscher (arxiv)

Hyperbolic Problems: Theory, Numerics, Applications (2018): 336.

To observe the dynamic formation of black holes in general relativity, one essentially needs to prove that closed trapped surfaces form during evolution from initial data that do not already contain trapped surfaces. We discuss the recent development of the construction of such admissible initial data for matter models. In addition, we extend known results for the Einstein equations coupled to perfect fluids in spherical symmetry and with linear equation of state to unbounded domains. Polytropic equations of state and regularity issues with the direct application of the singularity theorems in general relativity are discussed briefly.   Suggests a new stability problem that could be studied using SWIF convergence.

“Stability of graphical tori with almost nonnegative scalar curvature”

by Armando J. Cabrera Pacheco, Christian Ketterer, Raquel Perales (arxiv)

Calculus of Variations and Partial Differential Equations 59 Article 134 (2020).

Stability of graphical tori with almost nonnegative scalar curvature by Armando J. Cabrera Pacheco, Christian Ketterer, Raquel Perales By works of Schoen--Yau and Gromov--Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to 0. We prove flat and intrinsic flat subconvergence to a flat torus for sequences of 3-dimensional tori Mj that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound, a uniform lower bound on the area of the smallest closed minimal surface, and almost nonnegative scalar curvature.  We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space.

“Lower semicontinuity of ADM mass under intrinsic flat convergence"

by Jeffrey L. Jauregui, Dan A. Lee (arxiv)

Calculus of Variations & Partial Differential Equations 60.5 (2021).

A natural question in mathematical general relativity is how the ADM mass behaves as a functional on the space of asymptotically flat 3-manifolds of nonnegative scalar curvature. In previous results, lower semicontinuity has been established by the first-named author for pointed C2 convergence, and more generally by both authors for pointed C0 convergence (all in the Cheeger--Gromov sense). In this paper, we show this behavior persists for the much weaker notion of pointed Sormani--Wenger intrinsic flat volume (VF) convergence, under natural hypotheses. We consider smooth manifolds converging to asymptotically flat local integral current spaces (a new definition), using Huisken's isoperimetric mass as a replacement for the ADM mass. Along the way we prove results of independent interest about convergence of subregions of VF-converging sequences of integral current spaces. 

“Stability of the Spacetime Positive Mass Theorem in Spherical Symmetry”

by Edward Bryden, Marcus Khuri, and Christina Sormani  (arxiv)

JGEA Vol 31 Issue 4 4191-4239 (reprint) (DMS1612049)(SCGP)

The rigidity statement of the positive mass theorem asserts that an asymptotically flat initial data set for the Einstein equations with zero ADM mass, and satisfying the dominant energy condition, must arise from an embedding into Minkowski space. In this paper we address the question of what happens when the mass is merely small. In particular, we formulate a conjecture for the stability statement associated with the spacetime version of the positive mass theorem, and give examples to show how it is basically sharp if true. This conjecture is then established under the assumption of spherical symmetry in all dimensions. More precisely, it is shown that a sequence of asymptotically flat initial data satisfying the dominant energy condition, without horizons except possibly at an inner boundary, and with ADM masses tending to zero must arise from isometric embeddings into a sequence of static spacetimes converging to Minkowski space in the pointed volume preserving intrinsic flat sense. The difference of second fundamental forms coming from the embeddings and initial data must converge to zero in Lp, 1≤p<2. In addition some minor tangential results are also given, including the spacetime version of the Penrose inequality with rigidity statement in all dimensions for spherically symmetric initial data, as well as symmetry inheritance properties for outermost apparent horizons. 

“Properties of the Null Distance and Spacetime Convergence”

by Brian Allen and Annegret Burtscher (DMS1612049)(SCGP) (arxiv)

International Mathematics Research Notices 2022.10 (2022): 7729-7808.

The notion of null distance for Lorentzian manifolds recently introduced by Sormani and Vega gives rise to a metric and induces the manifold topology under mild assumptions on the time function of the spacetime. We show that, endowed with a (locally) anti-Lipschitz time function, the null distance is an intrinsic metric. In the case of completeness the anti-Lipschitz property of the time function is equivalent to the definiteness as well as the causality-encoding property of the time function. Furthermore, we prove that warped products of low regularity and globally hyperbolic spacetimes with complete Cauchy surfaces endowed with the null distance are integral current spaces. This metric and integral current structure sets the stage for investigating spacetime convergence analogous to Riemannian geometry. Our main theorem is a general Lorentzian convergence result for warped product spacetimes relating uniform convergence of warping functions to uniform, Gromov--Hausdorff and Sormani--Wenger intrinsic flat convergence of the corresponding null distances. In addition, we show that non-uniform convergence of warping functions in general leads to distinct limiting behavior, such as the convergence of null distances to Gromov--Hausdorff and Sormani--Wenger intrinsic flat limits that disagree.

``Sequences of three dimensional Manifolds with Positive Scalar Curvature"

by J Basilio and C Sormani  (arxiv) (NSF DMS 1612049)

Differential Geometry and its Applications 77 (2021): 101776.

We develop two new methods of constructing sequences of manifolds with positive scalar curvature that converge in the Gromov-Hausdorff and Intrinsic Flat sense to limit spaces with "pulled regions". The examples created rigorously within using these methods were announced by us a few years ago and have influenced the statements of some of Gromov's conjectures concerning sequences of manifolds with positive scalar curvature. Both methods extend the notion of "sewing along a curve" developed in prior work of the authors with Dodziuk to create limits that are pulled string spaces. The first method allows us to sew any compact set in a fixed initial manifold to create a limit space in which that compact set has been scrunched to a single point. The second method allows us to edit a sequence of regions or curves in a sequence of distinct manifolds.

"Relating Notions of Convergence in Geometric Analysis"

by B Allen and C Sormani  (arxiv) (NSF DMS 1612049)

Nonlinear Analysis 200 (2020): 111993.

We relate Lebesgue convergence of metric tensors or volume convergence to a given smooth metric to Intrinsic Flat and Gromov-Hausdorff convergence for sequences of Riemannian manifolds. We present many examples of sequences of conformal metrics which demonstrate that these notions of convergence do not agree in general even when the sequence is conformal to a fixed manifold. We then prove a theorem demonstrating that when sequences of metric tensors on a fixed manifold are bounded, and either the volumes converge or the metric tensors converge in the Lebesgue sense, then the Riemannian manifolds converge in the measured Gromov-Hausdorff and volume preserving Intrinsic Flat sense.

Volume Above Distance Below'

by Brian Allen, Raquel Perales, and Christina Sormani, (arxiv) (DMS1612049)(IAS)

to appear in Journal of Differential Geometry

Given a pair of metric tensors g1≥g0 on a Riemannian manifold, M, it is well known that Vol1(M)≥Vol0(M). Furthermore one has rigidity: the volumes are equal if and only if the metric tensors are the same g1=g0. Here we prove the that if gj≥g0 and Volj(M)→Vol0(M) then (M,gj) converge to (M,g0) in the volume preserving intrinsic flat sense. Well known examples demonstrate that one need not obtain smooth, C0, Lipschitz, or even Gromov-Hausdorff convergence in this setting. To complete our proof, we provide a new way of estimating the intrinsic flat distance between Riemannian manifolds.

Intrinsic Flat Stability of Manifolds with Boundary where Volume Converges and Distance is Bounded Below

by Brian Allen and  Raquel Perales (arxiv)

Given a compact, connected, and oriented manifold with boundary M and a sequence of smooth Riemannian metrics defined on it, gj, we prove volume preserving intrinsic flat convergence of the sequence to the smooth Riemannian metric g0 provided gj always mea- sures vectors strictly larger than or equal to g0, the diameter of gj is uniformly bounded, the volume of gj converges to the volume of g0, and L^{(m-1)/2} convergence of the metrics restricted to the boundary. Many examples are reviewed which justify and explain the intuition behind these hypotheses. These examples also show that uniform, Lipschitz, and Gromov-Hausdorff convergence are not appropriate in this setting. Our results provide a new rigorous method of proving some special cases of the intrinsic flat stability of the positive mass theorem.

Intrinsic flat convergence of points an applications to stability of the positive mass theorem

by Dan A. Lee, Lan-Hsuan Huang, and Raquel Perales (arxiv) 

Annales Henri Poincaré. Vol. 23. No. 7. Cham: Springer International Publishing, 2022.

Because of an error in the proof of Theorem 1.3 of Huang-Lee-Sormani, we provide an alternative proof that depends on recent results of Allen-Perales, in contrast to the use of Wenger's compactness theorem in the original proof. We also take the opportunity to flesh out some of the other arguments that were used in Huang-Lee-Sormani by proving some useful facts about point convergence in the Gromov-Hausdorff and intrinsic flat senses. In doing so, we clarify that all of the results of Huang-Lee-Sormani are true.    Comments: Sormani did not have the time to contribute to the addendum while organizing VWRS 2020 and has expressed that she is grateful for all their work on this.  Sormani was especially happy to see this paper, as she was unable to correct the original approach without developing a deep new understanding of the limits obtained under Wenger's Compactness Theorem. 

Almost non-negative scalar curvature on Riemannian manifolds conformal to tori

by Brian Allen (arxiv)

Journal of Geometric Analysis 31.11 (2021): 11190-11213.

In this article we reduce the geometric stability conjecture for the scalar torus rigidity theorem to the conformal case via the Yamabe problem. Then we are able to prove the case where a sequence of Riemannian manifolds is conformal to a uniformly controlled sequence of flat tori and satisfies the geometric stability conjecture. We are also able to handle the case where a sequence of Riemannian manifolds is conformal to a sequence of constant negative scalar curvature Riemannian manifolds which converge to a flat torus in C^1. The full conjecture from the conformal perspective is also discussed as a possible approach to resolving the conjecture.  

d_p convergence and epsilon regularity theorems for entropy and scalar curvature lower bounds

by Man-Chun Lee, Aaron Naber, and Robin Neumayer (arxiv)

Comments: In this article they introduce d_p which replaces the standard distance function d defined using lengths of curves.  They consider mGH convergence with respect to this d_p and show this works better than convergence with respect to the usual distance for questions involving lower scalar curvature and Perelman entropy bounds in all dimensions. Note the Perelman entropy bounds control the isoperimetric constant and prevent both bubbling and also prevent collapsing.  They prove scalar torus stability with these conditions with respect to d_p convergence and a compactness theorem for scalar curvature.   The proofs involve Ricci flow and Sobolev bounds on the metric tensors.  They define a new class of limit spaces which are not metric spaces but are rectifiable in a new sense of the word.  They have an example in 4D with scalar bounded below which shows GH and SWIF convergence is not correct in 4D under their hypotheses that is somewhat similar to Allen-Sormani examples.   These examples have closed geodesics of increasingly short length and unknown controls on MinA.   It would be very interesting to understand the value of d_p on the various examples in [SW-JDG] and in [BDS] and the Ilmanen Example.   It would be interesting to see if in 3D their results can be strengthened and how they compare to the 3D Scalar Torus Stability conjecture involving MinA.  It seems integral current spaces fit their definition of limit spaces but this is not definitely clear.   

Conjectures on Convergence and Scalar Curvature

by Christina Sormani (preliminary preprint) (arxiv preprint) 

Here we survey the compactness and geometric stability conjectures formulated by the participants at the 2018 IAS Emerging Topics Workshop on {\em Scalar Curvature and Convergence}.   We have tried to survey all the progress towards these conjectures as well as related examples, although it is impossible to cover everything.  We focus primarily on sequences of compact Riemannian manifolds with nonnegative scalar curvature and their limit spaces.  Christina Sormani is grateful to have had the opportunity to write up our ideas and has done her best to credit everyone involved within the paper even though she is the only author listed above.  In truth we are a team of over thirty people working together and apart on these deep questions and we welcome everyone who is interested in these conjectures to join us.   This is for a chapter in a book edited by Gromov and Lawson.

Conformal tori with almost non-negative scalar curvature

by Jianchun Chu and Man-Chun Lee (arxiv preprint)

In this work, we consider sequence of metrics with almost non-negative scalar curvature on torus. We show that if the sequence is uniformly conformal to another sequence of metrics with uniformly controlled geometry, then it converges to a flat metric in the volume preserving intrinsic flat sense, L^p sense and the measured Gromov-Hausdorff sense.   In the first theorem, the authors obtain intrinsic flat convergence using a uniform $L^p$ bound on the conformal factor, and in a separate theorem, with stronger hypotheses, they obtain mGH convergence.   They use Yamabe flow to complete their proof.   

Brill Lindquist Riemann Sums and their Limits

by Tatyana Benko and Iva Stavrov Allen (arxiv preprint)

This article commences a study of convergence of discretized point-object configurations, which we call Brill-Lindquist-Riemann sums, towards a charged dust continuum from the perspective of relativistic initial data. We are motivated by the interpretation of Brill-Lindquist-Riemann sums as collections of relatively isolated astrophysical bodies such as stars and galaxies in the universe, and the interpretation of the dust continuum as the universe itself. Our work begins by establishing the existence and the uniqueness of horizons/minimal surfaces of Brill-Lindquist metrics in the vicinity of the point-sources ("stars"). We then study the geometries of the regions exterior to said minimal surfaces, and discuss their Gromov-Hausdorff and intrinsic flat limit. An interesting and purely geometric byproduct of our work are examples in which the scalar curvature jumps upon taking Gromov-Hausdorff and /or intrinsic flat limits.   Comments: This includes an example of a sequence of manifolds with boundary that have nonnegative scalar curvature which converge in the intrinsic flat sense to a manifold which has strictly positive scalar curvature.   

From Varadhan's Limit to Eigenmaps: A Guide to the Geometric Analysis behind Manifold Learning

Chen-Yun Lin, Christina Sormani (arxiv) 

We present an overview of the history of the heat kernel and eigenfunctions on Riemannian manifolds and how the theory has lead to modern methods of analyzing high dimensional data via eigenmaps and other spectral embeddings. We begin with Varadhan's Theorem relating the heat kernel to the distance function on a Riemannian manifold. We then review various theorems which bound the heat kernel on classes of Riemannian manifolds. Next we turn to eigenfunctions, the Sturm-Liouville Decomposition of the heat kernel using eigenfunctions, and various theorems which bound eigenfunctions on classes of Riemannian manifolds. We review various notions of convergence of Riemannian manifolds and which classes of Riemannian manifolds are compact with respect to which notions of convergence. We then present Bérard-Besson-Gallot's heat kernel embeddings of Riemannian manifolds and the truncation of those embeddings. Finally we turn to Applications of Spectral embeddings to the Dimension Reduction of data sets lying in high dimensional spaces reviewing, in particular, the work of Belkin-Niyogi and Coifman-Lafon. We also review the Spectral Theory of Graphs and the work of Dodziuk and Chung and others. We close with recent theorems of Portegies and of the first author controlling truncated spectral embeddings uniformly on key classes of Riemannian manifolds. Throughout we provide many explicitly computed examples and graphics and attempt to provide as complete a set of references as possible. We hope that this article is accessible to both pure and applied mathematicians working in Geometric Analysis and their doctoral students.   Some of the open problems in this paper involve intrinsic flat convergence.

Stability of the positive mass theorem under Ricci curvature lower bounds

by Demetre Kazaras, Marcus Khuri, and Dan A. Lee (arxiv)

We establish Gromov-Hausdorff stability of the Riemannian positive mass theorem under the assumption of a Ricci curvature lower bound. .. In particular, this confirms Huisken and Ilmanen's conjecture on stability of the positive mass theorem under the assumptions described above. The proof is based on the harmonic level set approach to proving the positive mass theorem, combined with techniques used in the proof of Cheeger and Colding's almost splitting theorem.   Comments: They remark that Matveev-Portegies might then be applied to prove intrinsic flat convergence in this setting.   Note this would require extending  Matveev-Portegies to well defined regions in asymptotically flat manifolds as their theorem is only for manifolds without boundary.   It might be easier to apply one of the papers of Perales on the convergence of manifolds with boundary.

Semicontinuity of capacity under pointed intrinsic flat convergence

by Jeffrey L. Jauregui, Raquel Perales, Jacobus W. Portegies (arxiv)

The concept of the capacity of a compact set in Euclidean space generalizes readily to noncompact Riemannian manifolds and, with more substantial work, to metric spaces (where multiple natural definitions of capacity are possible). Motivated by analytic and geometric considerations, and in particular Jauregui's definition of capacity-volume mass and Jauregui and Lee's results on the lower semicontinuity of the ADM mass and Huisken's isoperimetric mass, we investigate how the capacity functional behaves when the background spaces vary. Specifically, we allow the background spaces to consist of a sequence of local integral current spaces converging in the pointed Sormani--Wenger intrinsic flat sense. For the case of volume-preserving convergence, we prove two theorems that demonstrate an upper semicontinuity phenomenon for the capacity: one version is for balls of a fixed radius centered about converging points; the other is for Lipschitz sublevel sets. Our approach is motivated by Portegies' investigation of the semicontinuity of eigenvalues under convergence. We include examples to show the semicontinuity may be strict, and that the volume-preserving hypothesis is necessary. Finally, there is a discussion on how capacity and our results may be used towards understanding the general relativistic total mass in non-smooth settings.

The spherical Plateau problem for group homology

by Antoine Song (arxiv)

Abstract: Given a group homology class, h,  of a countable group, Γ, we study a corresponding homological Plateau problem inside a canonical classifying space of Γ, which is defined using the regular representation of Γ and which is locally isometric to a Hilbert unit sphere. We investigate the relation between group theoretic properties of the pair (Γ,h) and the geometry of its Plateau solutions. For instance, we prove that for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M endowed with one third of its hyperbolic metric.

Characterizing candidates for Cannon's Conjecture from Geometric Measure Theory

Tamunonye Cheetham-West, Alexander Nolte (arxiv)

We show that recent work of Song implies that torsion-free hyperbolic groups with Gromov boundary are realized as fundamental groups of closed 3-manifolds of constant negative curvature if and only if the solution to an associated spherical Plateau problem for group homology is isometric to such a 3-manifold, and suggest some related questions.        

Lipschitz-volume rigidity of convex bodies among integral current spaces

G Basso, P Creutz, E Soultanis (arxiv)

Let $ X $ be an integral current space and $ C\subset\mathbb {R}^ n $ be a convex body. We show that if a $1 $-Lipschitz map $ f\colon X\to C $ of degree one does not increase volume and boundary volume, then it is an isometry. In other words, we prove that convex bodies enjoy a Lipschitz--volume rigidity property among integral current spaces. As an application we answer a question of Perales concerning the intrinsic flat convergence of minimizing sequences for the Plateau problem.  This paper has an interesting result which might be applied by those interested in proving intrinsic flat stability of the Positive Mass Theorem.  See the paper by Del Nin-Perales which proves a similar theorem and explains the application.  Warning: The first arxiv post of Basso-Creutz-Soultanis makes some strange statements in their introduction refering to a "Sormani-Wenger Compactness Theorem" (which does not exist) as a reason for filling volumes to be achieved.  See Portegies-Sormani "Properties..."  for more information about filling volumes using integral current spaces and Wenger "Compactness..." for better references.  The article has some troublesome notation using [[X]] in a different way than Ambrosio-Kirchheim and Sormani-Wenger.  

Rigidity of mass-preserving $1$-Lipschitz maps from integral current spaces into $\mathbb {R}^n$

G Del Nin, R Perales (arXiv:2210.06406)

We prove that given an n-dimensional integral current space and a 1-Lipschitz map from this space onto the n-dimensional Euclidean ball that preserves the mass of the current and is injective on the boundary then the map has to be an isometry. Then we show how to apply this result to prove the stability of the positive mass theorem for graphical manifolds as originally stated by Huang--Lee--Sormani.   This paper closes the gap in the original proof by Huang-Lee-Sormani and has many other exciting applications.  There is some overlap with the paper of Basso-Creutz-Soultanis but each paper has its own merits and deserves publication. The redundancy is the result of my having advertised the problem a few months earlier at a meeting in Oaxaca.   Del Nin contacted us with his interest and completed the work with Perales but we did not know anyone else was working on it.

Stability of the Spacetime Penrose Inequality in Spherical Symmetry

Emily Schaal (arXiv)

We formulate and prove the stability statement associated with the spacetime Penrose inequality for n-dimensional spherically symmetric, asymptotically flat initial data satisfying the dominant energy condition. We assume that the ADM mass is close to the half area radius of the outermost apparent horizon and, following the generalized Jang equation approach, show that the initial data must arise from an isometric embedding into a static spacetime close to to the exterior region of a Schwarzschild spacetime in the following sense. Namely, the time slice is close to the Schwarzschild time slice in the volume preserving intrinsic flat distance, the static potentials are close in L^2 loc, and the initial data extrinsic curvature is close to the second fundamental form of the embedding in L^2.   Comments: This is a key new stability result using intrinsic flat convergence.

Examples for Scalar Sphere Stability

Paul Sweeney (arXiv:2301:01292)

The rigidity theorems of Llarull and Marques-Neves, which show two different ways scalar curvature can characterize the sphere, have associated stability conjectures. Here we produce the first examples related to these stability conjectures. The first set of examples demonstrates the necessity of including a condition on the minimum area of all minimal surfaces to prevent bubbling along the sequence. The second set of examples constructs sequences that do not converge in the Gromov-Hausdorff sense but do converge in the volume preserving intrinsic flat sense. In order to construct such sequences, we improve the Gromov-Lawson tunnel construction so that one can attach wells and tunnels to a manifold with scalar curvature bounded below and only decrease the scalar curvature by an arbitrarily small amount. Moreover, we are able to generalize both the sewing construction of Basilio, Dodziuk, and Sormani, and the construction due to Basilio, Kazaras, and Sormani of an intrinsic flat limit with no geodesics.   Comments: This has key new sewing examples demonstrating intrinsic flat convergence with upper and lower bounds on the scalar curvature of the sequence.

Stability of Convex Disks

H Stufflebeam (arXiv:2301.13130)

We prove that topological disks with positive curvature and strictly convex boundary of large length are close to round spherical caps of constant boundary curvature in the Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat senses. This proves stability for a theorem of F. Hang and X. Wang. As an intermediate step we obtain a result concerning gauge fixing and compactness for solutions of a Liouville type PDE.    Comments: This paper applies Perales' Inner Region Theorem to show the SWIF and GH limits of these manifolds with boundary agree.   This is the first application in this direction. 

Geometric and Analytic Structures on Metric Spaces Homeomorphic to a Manifold

Giuliano Basso, Denis Marti, and Stefan Wenger (arxiv)

Abstract. We study metric spaces homeomorphic to a closed, oriented manifold from both geometric and analytic perspectives. We show that such spaces (which are sometimes called metric manifolds) admit a non-trivial integral current without boundary, provided they satisfy some weak assumptions. The existence of such an object should be thought of as an analytic analog of the fundamental class of the space and can also be interpreted as giving a way to make sense of Stokes’ theorem in this setting. We use this to establish (relative) isoperimetric inequalities in metric n-manifolds that are Ahlfors n-regular and linearly locally contractible. As an appli- cation, we obtain a short and conceptually simple proof of a deep theorem of Semmes about the validity of Poincare ́ inequalities in these spaces. In the smooth case the idea for such a proof goes back to Gromov. We also give sufficient conditions for a metric manifold to be rectifiable.

Intrinsic flat stability of the positive mass theorem for asymptotically hyperbolic graphical manifolds 

A J Cabrera Pacheco, M Graf, and R Perales (arXiv)

General Relativity and Gravitation, Vol 55, Issue 132, (2023) (reprint)

The rigidity of the Riemannian positive mass theorem for asymptotically hyperbolic manifolds states that the total mass of such a manifold is zero if and only if the manifold is isometric to the hyperbolic space. This leads to study the stability of this statement, that is, if the total mass of an asymptotically hyperbolic manifold is almost zero, is this manifold close to the hyperbolic space in any way? Motivated by the work of Huang, Lee and Sormani for asymptotically flat graphical manifolds with respect to intrinsic flat distance, we show the intrinsic flat stability of the positive mass theorem for a class of asymptotically hyperbolic graphical manifolds by adapting the positive answer to this question provided by Huang, Lee and the third named author.    Comments: This paper applies the Volume Above Distance Below Theorem of Allen-Perales-Sormani.

Entropy and stability of hyperbolic manifolds

Antoine Song (arxiv)

Let M be a closed oriented hyperbolic manifold of dimension at least 3 and let g0be the hyperbolic metric on M. Besson-Courtois-Gallot proved that for any Riemannian metric g on M with same volume as g0 , its volume entropy h(g) satisfies h(g)≥n−1with equality only when g is isometric to g0. We show that the hyperbolic metric is stable in the following sense. If gi is a sequence of Riemaniann metrics on M of same volume as g0 such that h(gi) converges to n−1, then (M,gi) converges to (M,g0) as metric measure spaces in the Gromov-Prokhorov topology. The proof is based on an intrinsic uniqueness result for spherical Plateau solutions of M.  Comments: This paper applies intrinsic flat convergence and Wenger’s Compactness Theorem to prove the main theorem.

Stability of Euclidean 3-space for the positive mass theorem

Conghan Dong, Antoine Song (arxiv)

We show that the Euclidean 3-space is stable for the Positive Mass Theorem in the following sense. Given a sequence of asymptotically flat 3-manifolds with nonnegative scalar curvature and suppose that their ADM masses converge to 0. Then for all i, there is a subset Zi in M3i such that the area of the boundary of Zi converges to zero and the sequence of pointed manifolds with Zi removed converges to in the pointed measured Gromov-Hausdorff topology. This confirms a conjecture of G. Huisken and T. Ilmanen.  Comments: This paper may be helpful towards proving intrinsic flat convergence in this general setting.

An Extreme Limit with Nonnegative Scalar Curvature

C Sormani, W Tian, C Wang - (arxiv)

In 2014, Gromov vaguely conjectured that a sequence of manifolds with nonnegative scalar curvature should have a subsequence which converges in some weak sense to a limit space with some generalized notion of nonnegative scalar curvature. The conjecture has been made precise at an IAS Emerging Topics meeting: requiring that the sequence be three dimensional with uniform upper bounds on diameter and volume, and a positive uniform lower bound on MinA, which is the minimum area of a closed minimal surface in the manifold. Here we present a sequence of warped product manifolds with warped circles over standard spheres, that have circular fibres over the poles whose length diverges to infinity, that satisfy the hypotheses of this IAS conjecture. We prove this sequence converges in the W1,p sense for p < 2 to an extreme limit space that has nonnegative scalar curvature in the distributional sense as defined by Lee-LeFloch and that the total distributional scalar curvature converges. This paper only requires expertise in smooth Riemannian Geometry, smooth minimal surfaces, and Sobolev Spaces. In a second paper, requiring expertise in metric geometry, the first two authors prove intrinsic flat and Gromov-Hausdorff convergence of our sequence to this extreme limit space and investigate its geometric properties.   Comments: The second paper concerning the intrinsic flat convergence has not yet appeared so this is worth keeping here for now.  

Finsler Currents 

Guiliano Basso (arxiv)

We propose a slight variant of Ambrosio and Kirchheim's definition of a metric current. We show that with this new definition it is possible to obtain certain volume functionals from Finsler geometry as mass measures of currents. As an application, we obtain a whole family of extendibly convex n-volume densities. This family includes the mass* and the circumscribed Riemannian volume densities.   Comments: These ideas should be useful when studying integral current spaces that are Finsler Manifolds.

On the Stability of Llarull's Theorem in Dimension Three 

Brian Allen, Edward Bryden, Demetre Kazaras (arxiv)

Llarull's Theorem states that any Riemannian metric on the n-sphere which has scalar curvature greater than or equal to n(n-1), and whose distance function is bounded below by the unit sphere's, is isometric to the unit sphere. Gromov later posed the {\emph{Spherical Stability Problem}}, which probes the flexibility of this fact. We give a resolution to this problem in dimension 3.  Informally, the main result asserts that a sequence of Riemannian 3-spheres whose distance functions are bounded below by the unit sphere's with uniformly bounded Cheeger isoperimetric constant and scalar curvatures tending to 6 must approach the round 3-sphere in the volume preserving Sormani-Wenger Intrinsic Flat sense. The argument is based on a proof of Llarull's Theorem due to Hirsch-Kazaras-Khuri-Zhang using spacetime harmonic functions.   Comments: The authors also apply the Volume Above Distance Below Theorem of Allen-Perales-Sormani.

Drawstrings and flexibility in the Georch conjecture

Demetre Kazaras and Kai Xu (arxiv)

In this paper, we observe new phenomena related to the structure of 3-manifolds satisfying lower scalar curvature bounds. We construct warped-product manifolds of almost nonnegative scalar curvature that converge to pulled string spaces in the Sormani-Wenger intrinsic flat topology. These examples extend the results of Lee-Naber-Neumayer to the case of dimension $3$. As a consequence, we produce the first counterexample to a conjecture of Sormani on the stability of the Geroch Conjecture. Our example tests the appropriate hypothesis for a related conjecture of Gromov. On the other hand, we demonstrate a $W^{1,p}$-stability statement ($1\leq p<2$) for the Geroch Conjecture in the class of warped products.   Comments: If this new preprint is correct, it may lead to a new class of examples of sequences of 3-manifolds with scalar curvature bounds that converge in the GH and the SWIF sense to pulled string spaces without introducing the small tunnels, small minimal surfaces, and increasing topology appearing in the sewing constructions of Dodziuk-Basilio-Sormani, Basilio-Sormani, and Sweeney.  Unlike the sewing, which can be done along any curve including those with endpoints in a manifold, this construction seems to require that the editing be done around a geodesic without endpoints.

Stability of Llarull's theorem in all dimensions

Sven Hirsch and Yiyue Zhang (arxiv)

We prove stability of Llarull's theorem in all dimensions using spin geometry. Our results are stated in terms of both intrinsic flat convergence and $C^0$ convergence outside a small set.   Comments: This assumes a Poincare Constant bound rather than the Cheeger Constant bound assumed by Allen-Bryden-Kazaras and uses Spin Geometry rather than harmonic functions.   To obtain intrinsic flat convergence, they prove volume convergence and then apply the Allen-Perales-Sormani Volume Above Distance Below Theorem.

On the Stability of the Yamabe Invariant of S^3

Liam Mazurowski and Xuan Yao (arxiv)

Abstract. Let g be a complete, asymptotically flat metric on R3 with vanishing scalar curvature. Moreover, assume that (R3, g) supports a nearly Euclidean L2 Sobolev inequality. We prove that (R3,g) must be close to Euclidean space with respect to the dp-distance defined by Lee-Naber-Neumayer. We then discuss some consequences for the stability of the Yamabe invariant of S3. More precisely, we show that if such a manifold (R3,g) carries a suitably normalized, positive solution to ∆g w + λw5 = 0 then w must be close, in a certain sense, to a conformal factor that transforms Euclidean space into a round sphere.   Comments: This does not have results directly involving intrinsic flat convergence but does discuss the notion as used for stability questions and has two stability results that might be useful towards proving intrinsic flat stability of the Yamabe Invariant.   So this is worth keeping here for now.  

Stability for the 3D Riemannian Penrose inequality

Conghan Dong (arxiv)

Abstract: We show that Schwarzschild manifold is stable for the 3-dimensional Riemannian Penrose inequality in the pointed measured Gromov-Hausdorff topology modulo negligible spikes and boundary area perturbations.   Comments: This does not use intrinsic flat convergence but has techniques which might be useful towards proving intrinsic flat stability for the Penrose Inequality as proposed in Lee-Sormani's Penrose paper listed above. So this is worth keeping here for now.  

Hyperbolic groups and spherical minimal surfaces

Antoine Song (arxiv)

Abstract: Let M be a closed, oriented, negatively curved, n-dimensional manifold with fundamental group Γ. Let S∞ be the unit sphere in ℓ2(Γ), on which Γ acts by the regular representation. The spherical volume of M is a topological invariant introduced by Besson-Courtois-Gallot. We show that it is equal to the area of an n-dimensional area-minimizing minimal surface inside the ultralimit of S∞/Γ, in the sense of Ambrosio-Kirchheim. Our proof combines the theory of metric currents with a study of limits of the regular representation of torsion-free hyperbolic groups.    Comments: This paper has techniques useful for intrinsic flat convergence.  See in particular the Appendix which extends the embedding theorem of SW-JDG.   So this is worth keeping here for now.  

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