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 $p < \frac{m}{2}$ and show a Sobolev inequality exists where an $L^{\frac{p}{2}}$ 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.