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 \(p < \frac{m}{2}\) where we show a Sobolev inequality exists between a Riemannian metric and its distance function. In particular, we show that 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.