Sobolev bounds and convergence of Riemannian manifolds

TL;DR

This paper establishes Sobolev bounds on Riemannian manifolds leading to uniform Hölder bounds on their distance functions, and explores convergence in Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat senses.

Contribution

It introduces a general trace inequality on Riemannian manifolds and links Sobolev bounds to manifold convergence in multiple geometric senses.

Findings

Uniform Sobolev bounds imply Hölder bounds on distance functions.

Sequences with volume bounds converge in Gromov-Hausdorff sense.

Additional volume bounds ensure convergence in Sormani-Wenger Intrinsic Flat sense.

Abstract

We consider sequences of compact Riemannian manifolds with uniform Sobolev bounds on their metric tensors, and prove that their distance functions are uniformly bounded in the H\"{o}lder sense. This is done by establishing a general trace inequality on Riemannian manifolds which is an interesting result on its own. We provide examples demonstrating how each of our hypotheses are necessary. In the Appendix by the first author with Christina Sormani, we prove that sequences of compact integral current spaces without boundary (including Riemannian manifolds) that have uniform H\"{o}lder bounds on their distance functions have subsequences converging in the Gromov--Hausdorff (GH) sense. If in addition they have a uniform upper bound on mass (volume) then they converge in the Sormani--Wenger Intrinsic Flat (SWIF) sense to a limit whose metric completion is the GH limit. We provide an example of a sequence developing a cusp demonstrating why the SWIF and GH limits may not agree.