Sobolev inequalities and convergence for Riemannian metrics and distance functions.

If one thinks of a Riemannian metric, g 1 , analogously as the gradient of the corresponding distance function, d 1 , with respect to a background Riemannian metric, g 0 , 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. [ABSTRACT FROM AUTHOR]