Isoperimetric and Sobolev Constant Estimate for Integral Ricci Curvature

Abstract: Integral curvature bound is a much weaker condition than the pointwise curvature bound and is a very natural notion as it occurs in diverse situations. We begin with introduction of the integral curvature and a review of early joint work with P. Petersen on the Laplacian and volume comparison for manifolds with only integral Ricci curvature bounds. We then present recent joint work with Xianzhe Dai and Zhenlei Zhang producing isoperimetric and Sobolev constant estimates for such manifolds without assuming a lower bound on volume. These estimates have many applications, in particular we obtain a maximum principle, a gradient estimate, and extend the L_2 Hessian estimate of Cheeger-Colding and Colding-Naber to manifolds with only lower bounds on their integral Ricci curvature.