The spectrum of the Laplacian on closed manifolds and the heat asymptotics near conical points

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=dx²+x²g(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.