Papers on the Intrinsic Flat Distance between Riemannian Manifolds
link to a presentation with graphics
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Acta Math. 185 (2000), no. 1, 1--80. (preprint, reprint, mathscinet review)
Abstract: We develop a theory of currents in metric spaces which extends the classical theory of Federer--Fleming in Euclidean spaces and in Riemannian manifolds. The main idea is to replace the duality with differential forms with the duality with (k + 1)-ples (f; 1 ; : : : ; k ) of Lipschitz functions, where k is the dimension of the current. We show, by a metric proof which is new even for currents in euclidean spaces, that the closure theorem and the boundary rectifiability theorem for integral currents hold in any complete metric space E. Moreover, we prove some existence results for a generalized Plateau problem in compact metric spaces and in some classes of Banach spaces, not necessarily finite dimensional.
There is also a Compactness Theorem and a Slicing Theorem in this paper. Note that this paper addresses weak convergence rather than flat convergence. Flat convergence is developed in "Flat Convergence for Integral Currents" by Stefan Wenger and is shown to be equivalent to weak convergence under some additional conditions. Key aspects of Ambrosio-Kirchheim's paper and prior work of Wenger are reviewed in the next paper.
Journal of Differential Geometry, Vol 87, (2011) (arxiv preprint)
Abstract (Dec 2008): Inspired by the Gromov-Hausdorff 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 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 H^{m} rectifiable spaces and give numerous examples.
"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."
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. 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 H^{m} 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 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, 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:
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.