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.

Note that 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 (preprint) rather than Ambrosio - Kirchheim's theory to handle currents of inifnite 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. The difficulty is that integral current spaces defined using Ambrosio-Kirchheim have finite volume. So one would need to define the limit space as a metric space with a point and an exhaustion by balls which are integral current spaces. Alternatively, one might try to apply the Lang-Wenger paper to define an intrinsic pointed convergence using Lang's thoery of local currents in metric spaces.

by Dan Lee and Christina Sormani

To appear in Crelle's Journal. (arxiv preprint)

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 1537-1556. (arxiv preprint)

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, as discussed in [LS]

by Sajjad Lakzian and Christina Sormani

Communications in Analysis and Geometry, Volume 21 (2013) No 1, 39-104 (arxiv preprint)

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.

by Sajjad Lakzian (arxiv preprint)

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 $H^{m-1}(S)=0$.

by Sajjad Lakzian (arxiv preprint)

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.

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

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.

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)

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.

by Christina Sormani

In progress: adding more properties. It already includes a theorem stating that sequences of manifolds which converge in the intrinsic flat sense have subdomains which converge in the Gromov-Hausdorff sense. It has a notion called sliced filling volumes of balls and another called the tetrhedral property, and sequences of spaces with approapriate uniform lower bounds on these properties and an upper bound on volume have subsequences which converge in the Gromov-Hausdorff and Intrinsic Flat sense. There is also an Arzela-Ascoli Theorem. On Feb 15 2012, this preliminary preprint was distributed including proofs of these results. Interval sliced filling and some additional compactness theorems were added in August 2012 update 50pp. On January 26, 2013, a new version was posted on the arxiv with new Arzela-Ascoli and Bolzano-Weierstrass type theorems some reorganization and clarifications (described in detail on the first page) and is here: preprint v2. More theorems and their proofs will be added in version 3.

More revisions to come as part of this paper has been moved to the Arzela Ascoli paper mentioned below.

by Misha Gromov, ihes preprint

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.

In this paper Gromov suggests the intrinsic flat distance might be useful to study sequences of manifolds with nonnegative scalar curvature, scal(g)>cost.

by Philippe LeFloch and Christina Sormani, arxiv preprint 38 pages (submitted)

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,

• the area of the boundary is continuous,
• the volume of the region is continuous and
• the Hawking mass of the boundary is continuous.

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.

by Christina Sormani, arxiv preprint

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.

by Raquel Perales, arxiv preprint (submitted)

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.

by Michael Munn, arxiv preprint (under revision)

Abstract: In prior papers, 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. In that paper, it is conjectured that this fact holds more generally, assuming a uniform lower Ricci curvature bound. Here we prove this conjecture. The original argument of Sormani-Wenger relies on careful estimates of Gromov's FillVol to control the density of regular points in the limit. Our proof takes a different approach and avoids the use the filling volume entirely. In doing so, our simplified proof generalizes their non-cancellation theorem to sequences with a uniform lower Ricci curvature bound. We do not handle manifolds with boundary.