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

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