From $$L^p$$ bounds to Gromov–Hausdorff convergence of Riemannian manifolds

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.