Quantitative Sobolev Extensions and the Neumann Heat Kernel for Integral Ricci Curvature Conditions

Abstract We prove the existence of Sobolev extension operators for certain uniform classes domains in a Riemannian manifold with an explicit bound on norm depending only geometry near their boundaries. use this quantitative estimate to obtain Neumann heat kernel upper bounds and gradient estimates positive solutions equation assuming integral Ricci curvature conditions geometric possibly non-convex boundary. Those also imply lower first eigenvalue considered domains.