Geodesic
Can one observe the bottleneckness of a space by the heat distribution?
Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 20 (2017) no. 3, pp. 77-88

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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 $\delta$-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 $\delta$-skin of the partition is also obtained.
Keywords: heat kernel, manifold with ends, inverse problem. 
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     author = {S. Ishiwata},
     title = {Can one observe the bottleneckness of a space by the heat distribution?},
     journal = {Matemati\v{c}eska\^a fizika i kompʹ\^uternoe modelirovanie},
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S. Ishiwata. Can one observe the bottleneckness of a space by the heat distribution?. Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 20 (2017) no. 3, pp. 77-88. http://geodesic.mathdoc.fr/item/VVGUM_2017_20_3_a6/