Weak convergence of currents and cancellation

Sormani, Christina; Wenger, Stefan

doi:10.1007/s00526-009-0282-x

Cited by 31 publications

(92 citation statements)

References 23 publications

(63 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An example showing that X need not be countably 2-rectifiable is given in [42,Theorem A.1]. Corollary 1.5 also holds for compact metric spaces of any topological dimension n and finite Hausdorff n-measure with positive lower density almost everywhere.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Quasiconformal almost parametrizations of metric surfaces

Damaris¹,

Wenger²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We look for minimal conditions on a two-dimensional metric surface X of locally finite Hausdorff 2-measure under which X admits an (almost) parametrization with good geometric and analytic properties. Only assuming that X is locally geodesic, we show that Jordan domains in X of finite boundary length admit a quasiconformal almost parametrization. If X satisfies some further conditions then such an almost parametrization can be upgraded to a geometrically quasiconformal homeomorphism or a quasisymmetric homeomorphism. In particular, we recover Rajala's recent quasiconformal uniformization theorem in the special case that X is locally geodesic as well as Bonk-Kleiner's quasisymmetric uniformization theorem. On the way we establish the existence of Sobolev discs spanning a given Jordan curve in X under nearly minimal assumptions on X and prove the continuity of energy minimizers.

show abstract

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Quasiconformal almost parametrizations of metric surfaces

Damaris¹,

Wenger²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…With no loss of generality we can assume that there exists an orientation ω of (X, x, H n ) associated with ω i . The first compatibility result between the intrinsic flat convergence and the mGHconvergence is given in [SW10] by Sormani-Wenger for compact manifolds with nonnegative Ricci curvature. After that Munn proved in [Mun14] similar compatibility for compact manifolds with bounded Ricci curvature.…”

Section: Compatibility With Convergence Of Metric Currentsmentioning

confidence: 99%

Ricci curvature and orientability

Honda

2017

Calc. Var.

View full text Add to dashboard Cite

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 L 2 -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.

show abstract

“…The continuity of the filling volume with respect to the intrinsic flat convergence follows from the following theorem. This fact was first observed by Sormani-Wenger [26] building upon work by Wenger on flat convergence of integral currents in metric spaces [28]. Then, (A.5) FillVol(∂M ′ (r)) = FillVol(∂M ′ (1))r n .…”

Section: Appendix a Filling Volume Of Spheres In Euclidean Spacementioning

confidence: 75%

“…Since the mass measure of an integral current space is lower semicontinuous with respect to intrinsic flat distance [27], here we study the notion of filling volume of a current. See [26]. In Theorem A.4 we see that the filling volume of a n dimensional sphere of radius r in Euclidean space rescales as r n times the filling volume of the n dimensional sphere in Euclidean space of radius 1. where ∂M = ∂N means that there is an orientation preserving isometry between ∂M and ∂N.…”

Section: Appendix a Filling Volume Of Spheres In Euclidean Spacementioning

confidence: 96%

The Sormani–Wenger intrinsic flat convergence of Alexandrov spaces

Perales²

2018

J. Topol. Anal.

View full text Add to dashboard Cite

We study sequences of integral current spaces (X j , d j , T j ) such that the integral current structure T j has weight 1 and no boundary and, all (X j , d j ) are closed Alexandrov spaces with curvature uniformly bounded from below and diameter uniformly bounded from above. We prove that for such sequences either their limits collapse or the Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat limits agree. The latter is done showing that the lower n dimensional density of the mass measure at any regular point of the Gromov-Hausdorff limit space is positive by passing to a filling volume estimate.In an appendix we show that the filling volume of the standard n dimensional integral current space coming from an n dimensional sphere of radius r > 0 in Euclidean space equals r n times the filling volume of the n dimensional integral current space coming from the n dimensional sphere of radius 1.

show abstract

Weak convergence of currents and cancellation

Cited by 31 publications

References 23 publications

Quasiconformal almost parametrizations of metric surfaces

Quasiconformal almost parametrizations of metric surfaces

Ricci curvature and orientability

The Sormani–Wenger intrinsic flat convergence of Alexandrov spaces

Contact Info

Product

Resources

About