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, <sup>w</sup><sup>e obtain an upper bound of the capacity growth of </sup><sup>δ</sup><sup>-skin </sup><sup>o</sup><sup>f the partition. By this</sup> 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.