The spectrum of the Laplacian on closed manifolds and the heat asymptotics near conical points
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917538955264000 |
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| author | Roidos, Nikolaos |
| author_facet | Roidos, Nikolaos |
| contents | Let $\mathcal{M}$ be a smooth, closed and connected manifold of dimension $n\in\mathbb{N}$, endowed with a Riemannian metric $g$. Moreover, let $\mathcal{B}$ be an $(n+1)$-dimensional compact manifold with boundary equal to $\mathcal{M}$. Endow $\mathcal{B}$ with a Riemannian metric $h$ such that, in local coordinates $(x,y)\in [0,1)\times \mathcal{M}$ on the collar part of the boundary, it admits the warped product form $h=dx^{2}+x^{2}g(y)$. We consider the homogeneous heat equation on $(\mathcal{B},h)$ 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 $(\mathcal{M},g)$ determines explicitly the above asymptotic expansion and vice versa. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_00573 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The spectrum of the Laplacian on closed manifolds and the heat asymptotics near conical points Roidos, Nikolaos Analysis of PDEs Spectral Theory Primary: 35B40, 35C20, 35K05, 35K65, 35R01, 47A11, 58J50. Secondary: 35P15, 47A10, 47B12, 58C40 Let $\mathcal{M}$ be a smooth, closed and connected manifold of dimension $n\in\mathbb{N}$, endowed with a Riemannian metric $g$. Moreover, let $\mathcal{B}$ be an $(n+1)$-dimensional compact manifold with boundary equal to $\mathcal{M}$. Endow $\mathcal{B}$ with a Riemannian metric $h$ such that, in local coordinates $(x,y)\in [0,1)\times \mathcal{M}$ on the collar part of the boundary, it admits the warped product form $h=dx^{2}+x^{2}g(y)$. We consider the homogeneous heat equation on $(\mathcal{B},h)$ 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 $(\mathcal{M},g)$ determines explicitly the above asymptotic expansion and vice versa. |
| title | The spectrum of the Laplacian on closed manifolds and the heat asymptotics near conical points |
| topic | Analysis of PDEs Spectral Theory Primary: 35B40, 35C20, 35K05, 35K65, 35R01, 47A11, 58J50. Secondary: 35P15, 47A10, 47B12, 58C40 |
| url | https://arxiv.org/abs/2312.00573 |