Sobolev Inequalities and Convergence For Riemannian Metrics and Distance Functions

Abstract

If one thinks of a Riemannian metric, g1, analogously as the gradient of the corresponding distance function, d1, with respect to a background Riemannian metric, g0, then a natural question arises as to whether a corresponding theory of Sobolev inequalities exists between the Riemannian metric and its distance function. In this paper we study the sub-critical case p < m 2 where we show a Sobolev inequality exists between a Riemannian metric and its distance function. In particular, we show that an L p 2 bound on a Riemannian metric implies an L q bound on its corresponding distance function. We then use this result to state a convergence theorem and show how this theorem can be useful to prove geometric stability results by proving a version of Gromov's conjecture for tori with almost non-negative scalar curvature in the conformal case. Examples are given to show that the hypotheses of the main theorems are necessary.