Abstract Let M M be a smooth, closed and connected manifold of dimension n N n ∈ N, endowed with a Riemannian metric g. Moreover, let B B be an (n+1) (n + 1) -dimensional compact manifold with boundary equal to M M. Endow B B with a Riemannian metric h such that, in local coordinates (x, y) [0, 1) M (x, y) ∈ [ 0, 1) × M on the collar part of the boundary, it admits the warped product form h=dx^2+x^2g (y) h = d x 2 + x 2 g (y). We consider the homogeneous heat equation on (B, h) (B, h) and find an arbitrary long asymptotic expansion of the solutions with respect to x near 0. It turns out that the spectrum of the Laplacian on (M, g) (M, g) determines explicitly the above asymptotic expansion and vice versa.
Nikolaos Roidos (Fri,) studied this question.