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From \(L^p\) bounds to Gromov–Hausdorff convergence of Riemannian manifolds

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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 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 \(L^p\), \(p > \frac{m}{2}\) 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 can be achieved.

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Funding

This research was funded in part by NSF DMS - 1612049.

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  1. Lehman College, CUNY, Bronx, NY, USA

    Brian Allen

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  1. Brian Allen
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Correspondence to Brian Allen.

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Allen, B. From \(L^p\) bounds to Gromov–Hausdorff convergence of Riemannian manifolds. Geom Dedicata 218, 30 (2024). https://doi.org/10.1007/s10711-023-00875-y

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