Quantitative Sobolev extensions and the Neumann heat kernel for integral Ricci curvature conditions

TL;DR

This paper establishes Sobolev extension operators and heat kernel bounds for domains in Riemannian manifolds under integral Ricci curvature conditions, linking geometric properties to analytical estimates.

Contribution

It introduces explicit Sobolev extension bounds and heat kernel estimates under integral Ricci curvature assumptions, advancing geometric analysis techniques.

Findings

Existence of Sobolev extension operators with explicit bounds

Uniform Neumann heat kernel upper bounds under integral Ricci curvature

Quantitative lower bounds on the first Neumann eigenvalue

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 boundary. Those estimates also imply quantitative lower bounds on the first Neumann eigenvalue of the considered domains.