.

Papers on the Intrinsic Flat Distance between Riemannian Manifolds

ordered according to availability as a preprint with abstracts and comments

2009 Geometry Festival Presentation and other presentations

Anyone interested in working on SWIF convergence should email Sormani to ensure no one else is working on their planned project.


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

      Journal of Differential Geometry, Vol 87, (2011) (arxiv preprint) (early preprint distributed in Fall 2008) (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 dZ( [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"

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

      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 Appendix by Raanan Schul and Stefan Wenger

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

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

      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

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

      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

      Annales Henri Poincare November 2012, Volume 13, Issue 7, pp 1537-1556. (arxiv preprint)

      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 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.  

    [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 47August 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] propagates 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 DedicataDecember 2015, Volume 179, Issue 1pp 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 August 2014Volume 12Issue 8pp 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 Gromovihes preprint (arxiv preprint later)

      Central European Journal of Mathematics July 2014Volume 12Issue 7pp 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 preprintCommunications 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

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

      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,  

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

      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 Peralesarxiv 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 Munnarxiv preprint 

      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 for industry.

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

      by Lan-Hsuan HuangDan LeeChristina Sormaniarxiv 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: Cabrera-Pacheco and Perales questioned a step in the proof near the very end of the paper.  An erratum is in progress which provides a new proof of the main theorem as originally stated applying work of Allen-Perales that circumvents this step.   It is conjectured that the original proof holds but new theory on integral current spaces needs to be developed to justify the steps adequately.

    [SS16] "Intrinsic Flat Convergence of Covering Spaces"

      Geometriae Dedicata, Vol 184 (2016) pages 83–114  (arxiv preprint(reprintDOI 10.1007/s10711-016-0158-0

      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"


      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"


      Journal of Topology and Analysis (2018)

      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"

      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"

      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"


      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"


      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"


      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"
      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.
    [SS19] "Geometrostatic manifolds of small ADM Mass"


      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"

      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"

      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"
      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”

      This reports on current work with Sakovich and Vega.

              [CL]  ``Positive Scalar Curvature with Skeleton Singularities"
      Math Annalen 374, 99-131 (2019). 

      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.   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"


      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”
      to appear in Nonlinear Analysis
       
      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''

      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)

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


      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. 
    Stability of graphical tori with almost nonnegative scalar curvature”
      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"
      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”


      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 Lp1p<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”

      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 Manifolds with Positive Scalar Curvature"


      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"


      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)

      Given a pair of metric tensors g1g0 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 gjg0 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.






This page is maintained by Christina Sormani who has National Science Foundation funding to conduct research on the Intrinsic Flat Distance: DMS 1006059 and DMS 1309360 and DMS 1612409..  
The comments are by Christina Sormani.