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# Intrinsic Ultracontractivity for Domains in Negatively Curved Manifolds

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 Title: Intrinsic Ultracontractivity for Domains in Negatively Curved Manifolds Authors: Aikawa, Hiroaki Browse this author van den Berg, Michiel Browse this author Masamune, Jun Browse this author Keywords: Intrinsic ultracontractivity Ricci curvature First eigenvalue Heat kernel Torsion function Capacitary width Issue Date: 12-Oct-2021 Publisher: Springer Journal Title: Computational Methods and Function Theory Publisher DOI: 10.1007/s40315-021-00402-8 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 forwhich 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 Poincare inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale. Type: article URI: http://hdl.handle.net/2115/82920 Appears in Collections: 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)