Intrinsic Ultracontractivity for Domains in Negatively Curved Manifolds

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...

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Bibliographic Details
Published in:Computational methods and function theory Vol. 21; no. 4; pp. 797 - 824
Main Authors: Aikawa, Hiroaki, van den Berg, Michiel, Masamune, Jun
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2021
Springer Nature B.V
Subjects:
Analysis
Computational Mathematics and Numerical Analysis
Dirichlet problem
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Riemann manifold
31C12
First eigenvalue
Ricci curvature
Heat kernel
Capacitary width
Intrinsic ultracontractivity
58J35
Torsion function
31B15
ISSN:1617-9447, 2195-3724
Online Access:Get full text
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Summary: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 Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale.
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ISSN:1617-9447
2195-3724
DOI:10.1007/s40315-021-00402-8