Intrinsic ultracontractivity for domains in negatively curved manifolds

TL;DR

This paper establishes conditions under which the Dirichlet heat semigroup on certain domains in negatively curved manifolds exhibits intrinsic ultracontractivity, linking geometric and spectral properties through capacitary width.

Contribution

It provides a new sufficient condition for intrinsic ultracontractivity based on capacitary width in negatively curved manifolds, connecting spectral and geometric analysis.

Findings

Reciprocal of the spectral bottom is comparable to the square of capacitary width.

Supremum of the torsion function is comparable to the square of capacitary width.

Conditions involve volume doubling, Poincaré inequality, and Gaussian heat kernel estimates.

Abstract

Let $M$ be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets $D$ in $M$ for which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(D)$, and the supremum of the torsion function for $D$ are comparable with the square of the capacitary width for $D$ if the latter is sufficiently small. The technical key ingredients are the volume doubling property, the Poincar\'e inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel for finite scale.