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 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.

Integral Ricci curvature and the mass gap of Dirichlet Laplacians on domains

“…The desired Neumann eigenvalue estimate then follows from the main results in [12]. Proposition 1 also generalizes the recently appeared eigenvalue estimate in [19, Corollary 1.5] obtained by completely different techniques.…”

“…This paper provides a gap estimate for suitable subsets of manifolds assuming only integral bounds on the negative part of the Ricci curvature. Such integral curvature conditions gathered a lot of attention during the last decades because in contrast to lower Ricci curvature pointwise bounds, they are more stable under perturbations of the metric, see, for example, [2, 4, 5, 9, 11, 17–19, 21, 22, 24, 25] and the references therein.…”