  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.
