Ishiwata S. Can One Observe the Bottleneckness of a Space by the Heat Distribution

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https://doi.org/10.15688/mpcm.jvolsu.2017.3.6

Satoshi Ishiwata
Professor, Department of Mathematical Sciences,
Yamagata University
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Yamagata 990-8560, Japan

Abstract. In this paper we discuss a bottleneck structure of a non-compact manifold appearing in the behavior of the heat kernel. This is regarded as an inverse problem of heat kernel estimates on manifolds with ends obtained in [10] and [8]. As a result, if a non-parabolic manifold is divided into two domains by a partition and we have suitable heat kernel estimates between different domains, we obtain an upper bound of the capacity growth of δ-skin of the partition. By this estimate of the capacity, we obtain an upper bound of the first non-zero Neumann eigenvalue of Laplace — Beltrami operator on balls. Under the assumption of an isoperimetric inequality, an upper bound of the volume growth of the δ-skin of the partition is also obtained.

Key words: heat kernel, manifold with ends, inverse problem.

Can One Observe the Bottleneckness of a Space by the Heat Distribution by Ishiwata S. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Mathematical Physics and Computer Simulation. №3 (40) 2017 pp. 77-88

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