Sobolev inequalities and convergence for Riemannian metrics and distance functions

If one thinks of a Riemannian metric, $g_1$, analogously as the gradient corresponding distance function, $d_1$, with respect to background $g_0$, then natural question arises whether theory Sobolev inequalities exists between metric and its function. In this paper we study sub-critical case $p < \frac{m}{2}$ show inequality where an $L^{\frac{p}{2}}$ bound on implies $L^q$ We use result state convergence theorem how can be useful prove geometric stability results by proving version Gromov's conjecture for tori almost non-negative scalar curvature in conformal case. Examples are given that hypotheses main theorems necessary.