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
$\delta$-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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