From $L^p$ Bounds To Gromov-Hausdorff Convergence Of Riemannian Manifolds

TL;DR

This paper develops a method to convert $L^p$ bounds on Riemannian metrics into H"older bounds on distance functions, leading to new compactness, convergence, and stability results in Riemannian geometry.

Contribution

It introduces a Morrey-type inequality for Riemannian manifolds that links $L^p$ bounds to H"older control of the distance function, enabling new geometric stability results.

Findings

Establishes a new estimate converting $L^p$ bounds into H"older bounds on distances.

Proves a compactness theorem for Riemannian manifolds under these bounds.

Demonstrates convergence and scalar torus stability results.

Abstract

In this paper we provide a way of taking $L^p$, $p > \frac{m}{2}$ bounds on a $m-$ dimensional Riemannian metric and transforming that into H\"{o}lder bounds for the corresponding distance function. One can think of this new estimate as a type of Morrey inequality for Riemannian manifolds where one thinks of a Riemannian metric as the gradient of the corresponding distance function so that the $L^p$, $p > \frac{m}{2}$ bound analogously implies H\"{o}lder control on the distance function. This new estimate is then used to state a compactness theorem, another theorem which guarantees convergence to a particular Riemmanian manifold, and a new scalar torus stability result. We expect these results to be useful for proving geometric stability results in the presence of scalar curvature bounds when Gromov-Hausdorff convergence is expected.