Heat kernel asymptotics for real powers of Laplacians
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911869776691200 |
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| author | Anghel, Cipriana |
| author_facet | Anghel, Cipriana |
| contents | We describe the small-time heat kernel asymptotics of real powers $Δ^r$, $r \in (0,1)$ of a non-negative self-adjoint generalized Laplacian $Δ$ acting on the sections of a hermitian vector bundle $\mathcal E$ over a closed oriented manifold $M$. First we treat separately the asymptotic on the diagonal of $M \times M$ and in a compact set away from it. Logarithmic terms appear only if $n$ is odd and $r$ is rational with even denominator. We prove the non-triviality of the coefficients appearing in the diagonal asymptotics, and also the non-locality of some of the coefficients. In the special case $r=1/2$, we give a simultaneous formula by proving that the heat kernel of $Δ^{1/2}$ is a polyhomogeneous conormal section in $\mathcal E \boxtimes {\mathcal E}^* $ on the standard blow-up space $M_{heat}$ of the diagonal at time $t=0$ inside $[0,\infty)\times M \times M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_14142 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Heat kernel asymptotics for real powers of Laplacians Anghel, Cipriana Differential Geometry 58J37, 58J35 We describe the small-time heat kernel asymptotics of real powers $Δ^r$, $r \in (0,1)$ of a non-negative self-adjoint generalized Laplacian $Δ$ acting on the sections of a hermitian vector bundle $\mathcal E$ over a closed oriented manifold $M$. First we treat separately the asymptotic on the diagonal of $M \times M$ and in a compact set away from it. Logarithmic terms appear only if $n$ is odd and $r$ is rational with even denominator. We prove the non-triviality of the coefficients appearing in the diagonal asymptotics, and also the non-locality of some of the coefficients. In the special case $r=1/2$, we give a simultaneous formula by proving that the heat kernel of $Δ^{1/2}$ is a polyhomogeneous conormal section in $\mathcal E \boxtimes {\mathcal E}^* $ on the standard blow-up space $M_{heat}$ of the diagonal at time $t=0$ inside $[0,\infty)\times M \times M$. |
| title | Heat kernel asymptotics for real powers of Laplacians |
| topic | Differential Geometry 58J37, 58J35 |
| url | https://arxiv.org/abs/2203.14142 |