Intrinsic Ultracontractivity for Domains in Negatively Curved Manifolds

Hiroaki Aikawa, Michiel van den Berg, Jun Masamune

Research output: Contribution to journalArticlepeer-review

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 L2(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 Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale.

Original languageEnglish
Pages (from-to)797-824
Number of pages28
JournalComputational Methods and Function Theory
Volume21
Issue number4
DOIs
Publication statusPublished - 2021 Dec

Keywords

  • Capacitary width
  • First eigenvalue
  • Heat kernel
  • Intrinsic ultracontractivity
  • Ricci curvature
  • Torsion function

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