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

2021/06/27 by Brian Allen, Allen, Brian · 1 citation
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paper · pdf · doi:10.48550/arxiv.2106.14231

openalex publication_date 2021/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we provide a way of taking L^p, p > (m)/(2) bounds on a m- dimensional Riemannian metric and transforming that into Hölder 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 > (m)/(2) bound analogously implies Hölder 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.

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