April 9, 2018
TITLE: Manifolds with integral curvature bounds
ABSTRACT: We begin with 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 X. Dai and Z. Zhang producing a local Sobolev constant estimate for such manifolds without assuming a lower bound on volume. We close with applications of this theorem to produce a maximum principle, a gradient estimate, and to extend the Hessian estimate of Cheeger-Colding and Colding-Naber to manifolds with only lower bounds on their integral Ricci curvature.