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

2023/12/01 by Nikolaos Roidos, Roidos, Nikolaos
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2312.00573

Abstract

Let M be a smooth, closed and connected manifold of dimension n∈ℕ, endowed with a Riemannian metric g. Moreover, let B be an (n+1)-dimensional compact manifold with boundary equal to M. Endow B with a Riemannian metric h such that, in local coordinates (x,y)∈ [0,1)× M on the collar part of the boundary, it admits the warped product form h=dx²+x²g(y). We consider the homogeneous heat equation on (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) determines explicitly the above asymptotic expansion and vice versa.

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