"The Intrinsic Flat Distance between Riemannian Manifolds and Integral Current Spaces"by Christina Sormani and Stefan WengerJournal 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 d_{F}(M,N) = inf d^{Z}_{F }( [f(M)] , [h(N)] )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 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 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
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.
"Cancellation under flat convergence")
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 H "The pointed flat compactness theorem for locally integral currents"
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. "Stability of the Positive Mass Theorem for Rotationally Symmetric Riemannian Manifolds"
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. "Almost Equality in the Penrose Equality for Rotationally Symmetric Riemannian Manifolds"
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. "Smooth Convergence Away from Singular Sets"
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. "Diameter Controls and Smooth Convergence away from Singular Sets"
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 $H^{m-1}(S)=0$. ``Continuity of Ricci flow through Neck Pinch Singularities"
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. ``Dirac and Plateau Billiards in Domains with Corners"
Central European Journal of Mathematics August 2014, Volume 12, Issue 8, pp 1109-1156Abstract: 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. "Semicontinuity of eigenvalues under intrinsic flat convergence"
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. "Plateau Stein Manifolds"
Central European Journal of Mathematics July 2014, Volume 12, Issue 7, pp 923-951Abstract: We study/construct (proper and non-proper) 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 non-critical 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. "Intrinsic Flat Arzela-Ascoli Theorems"
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. - the depth of the region and
- the Hawking mass of the boundary,
- the area of the boundary is continuous,
- the volume of the region is continuous and
- the Hawking mass of the boundary is continuous.
"Nonlinear stability of rotationally symmetric spaces with low regularity"
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. "On Lower Semicontinuity of the ADM mass"
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. "Volumes and Limits of Manifolds with Ricci Curvature and Mean Curvature Bounds"
"Intrinsic Flat Convergence with bounded Ricci curvature"
"Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space"
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 Gromov-Hausdorff and intrinsic flat compactness theorem for sequences of metric spaces with uniform Lipschitz bounds on their metrics. "Intrinsic Flat Convergence of Covering Spaces"by Zahra Sinaei and Christina Sormani, Vol 184 (2016) pages 83–114 (arxiv preprint) (reprint) DOI 10.1007/s10711-016-0158-0Geometriae Dedicata,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.
arxiv preprintRaquel Perales 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. "Properties of the Intrinsic Flat Distance"by Jacobus Portegies and Christina Sormani, arxiv preprint, (completed in April 2015) 56 pages to appear in a special volume in honor of Y Burago edited by D Burago and S BuyaloAbstract: 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."Convergence of Manifolds and Metric Spaces with Boundary"by Raquel Perales, arxiv preprint, (completed in May 2015) 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"Intrinsic Flat and Gromov-Hausdorff Convergence for Manifolds with Ricci curvature bounded below"by Rostitslav Matveev and Jacobus Portegies, arxiv preprint, (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.
"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.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.
"Ricci Curvature and Orientability"by Shouhei Honda, arxiv preprint, to appear in CVPDEAbstract: 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. "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.
"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)
"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."Geometrostatic manifolds of small ADM Mass"by C Sormani and I Stavrov, 42 pages, (arxiv preprint) 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) "A generalized tetrahedral property for spaces with conical singularities"by P Nunez-Zimbron and R Perales (arxiv preprint)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)"Contrasting Various Notions of Convergence in Geometric Analysis"by Brian Allen and Christina Sormani, (figures by Pen Chang) 43 pages, (arxiv preprint)- 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)
“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; 21 pages, (arxiv preprint)to appear in Geometriae Dedicata-
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.
“Oberwolfach Report: 2012 Spacetime Intrinsic Flat Convergence” by C Sormani (arxiv) |