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.
Abstract: Inspired by the GromovHausdorff distance, we define the intrinsic flat distance between oriented m dimensional Riemannian manifolds with boundary as:
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 AmbrosioKirchheim, 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 H^{m} 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 FedererFlemming’s “Normal and Integral Currents” Annals of Math 2 72 (1960) 458520 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 AmbrosioKirchheim’s “Currents in Metric Spaces” Acta Math Vol 185 2000 no 1, 180, 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 AmbrosioKirchheim’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 GromovHausdorff 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. "Compactness for manifolds and integral currents with bounded diameter and volume"
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 AmbrosioKirchheim) 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. "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. 12, 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 GreenePetersen, we show that the flat limits and GromovHausdorff limits of linearly locally contractible manifolds agree. As a consequence the limits of these spaces are countably H^{m} rectifiable spaces. Applying CheegerColding and Perelman, we show that the flat limits and GromovHausdorff limits of noncollapsing sequences of manifolds with nonnegative Ricci curvature agree. CheegerColding 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 AmbrosioKirchheim. 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. by Urs Lang and Stefan Wenger Communications in Analysis and Geometry. 19 (2011), no. 1, 159189. (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 Ambrosioâ€“Kirchheim theory of currents in metric spaces, including currents with ï¬nite 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. 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. by Dan Lee and Christina Sormani Annales Henri Poincare November 2012, Volume 13, Issue 7, pp 15371556. (arxiv preprint) Abstract: This article is the sequel to our previous paper [LS] dealing with the nearequality case of the Positive Mass Theorem. We study the nearequality 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 nearequality results. by Sajjad Lakzian and Christina Sormani Communications in Analysis and Geometry, Volume 21 (2013) No 1, 39104 (arxiv preprint) 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 GromovHausdorff or Intrinsic Flat limit of the manifolds. Many examples are given. 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 nvolume in metric spaces with Besicovitchtype inequalities builtin 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., Plateauhedra, Scalar Curvature and Dirac Billiards, in preparation]. Comments: It just says the (1+epsilon) biLipschitz construction is too strong and upcoming work will consider the intrinsic flat distance. by Sajjad Lakzian (arxiv preprint) (reprint) Journal of Differential Geometry and Applications, Volume 47, August 2016, Pages 99129 Abstract: This paper builds upon and extends prior results of the author and C. Sormani considering both the Intrinsic Flat and GromovHausdorff limits of sequences of manifolds with metrics that converge smoothly away from singular sets $S\subset M^m$ of Hausdorff measure $H^{m1}(S)=0$. by Sajjad Lakzian (arxiv preprint) (reprint) Geometriae Dedicata, December 2015, Volume 179, Issue 1, pp 69–89 Abstract: In this article, we consider the AngenentCaputoKnopf's Ricci Flow through neckpinch singularities. We will explain how one can see the ACK'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 SormaniWenger Intrinsic Flat (SWIF) distance. by Misha Gromov (posted preprint, (Sept 30, 2013)) (arxiv preprint later) 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. by Jacobus W. Portegies (arxiv preprint) Calculus of Variations and Partial Differential Equations October 2015, Volume 54, Issue 2, pp 17251766 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 minmax values based on the normalized energy and show that when integral current spaces converge in the intrinsic flat sense without loss of volume, the minmax values of the limit space are larger than or equal to the upper limit of the minmax 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 Poincarelike inequality. by Misha Gromov, ihes preprint (arxiv preprint later) Abstract: We study/construct (proper and nonproper) Morse functions f on complete Riemannian manifolds $X$ such that the hypersurfaces f(x) = t for all âˆ’âˆž < t < +âˆž have positive mean curvatures at all noncritical points x âˆˆ 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. by Christina Sormani, arxiv preprint, Communications in Analysis and Geometry Vol. 27, No 1, 2019 Abstract: In this paper two ArzelaAscoli 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 BolzanoWeierstrass 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 GromovHausdorff limit of regions within those manifolds. Open problems with suggested applications are provided throughout the paper.
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: 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.
Communications in Analysis and Geometry Vol 26 issue 1 (2018) 18 pages arxiv preprint
by Raquel Perales, arxiv preprint Journal of Differential Geometry and Applications Vol 48, (2016) pp 2337 (reprint) by Michael Munn, arxiv preprint by LanHsuan Huang, Dan Lee, Christina Sormani, arxiv preprint to appear in Crelle (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 GromovHausdorff and intrinsic flat compactness theorem for sequences of metric spaces with uniform Lipschitz bounds on their metrics.
Abstract: We examine the limits of covering spaces and the covering spectra of oriented Riemannian manifolds, M_{j}, 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 S^{4} that converge in the intrinsic flat sense to a torus S^{1}×S^{3}. 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. "On the SormaniWenger Intrinsic Flat Convergence of Alexandrov Spaces" by Nan Li and Raquel Perales arxiv preprint 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 SormaniWenger Intrinsic Flat convergence and GromovHausdorff convergence agree.
“Oberwolfach Report: 2012 Spacetime Intrinsic Flat Convergence”
This reports on work with Sakovich and Vega. “Inverse Mean Curvature Flow and the Stability of the Positive Mass
Theorem”
function.
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 GromovHausdorff 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)
by Jiewon Park, Wenchuan Tian, Changliang Wang (arxiv) 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.
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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.
