Can one observe the bottleneckness of a space by the heat distribution?

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.