The spectrum of the Laplacian on closed manifolds and the heat asymptotics near conical points
Year: 2026
Type: article
Abstract: Let $$\mathcal {M}$$ M be a smooth, closed and connected manifold of dimension $$n\in \mathbb {N}$$ n ∈ N , endowed with a Riemannian metric g . Moreover, let $$\mathcal {B}$$ B be an $$(n+1)$$ ( n + 1 ) -dimensional compact manifold with boundary equal to $$\mathcal {M}$$ M . Endow $$\mathcal {B}$$ B with a Riemannian metric h such that, in local coordinates $$(x,y)\in [0,1)\times \mathcal {M}$$ ( x , y ) ∈ [ 0 , 1 ) × M on the collar part of the boundary, it admits the warped product form $$h=dx^{2}+x^{2}g(y)$$ h = d x 2 + x 2 g ( y ) . We consider the homogeneous heat equation on $$(\mathcal {B},h)$$ ( 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 $$(\mathcal {M},g)$$ ( M , g ) determines explicitly the above asymptotic expansion and vice versa.
Source: Semigroup Forum
Author: Nikolaos Roidos
Institution: University of Patras
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W7128813847