Heat coefficients of surfaces with curved conic singularities
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911310088765440 |
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| author | Schueth, Dorothee |
| author_facet | Schueth, Dorothee |
| contents | Let $(M,g)$ be a two-dimensional Riemannian manifold of finite diameter with a conical singularity. Under the assumption that the metric near the cone point $C$ is rotationally invariant, but not necessarily flat, we give an explicit formula for the coefficient $b_{1/2}(C)$ in the heat trace expansion $\operatorname{tr}(\operatorname{exp}(-tΔ_g))\sim_{t\searrow0} (4πt)^{-1}\sum_{j=0}^\infty a_j(M) t^j+\sum_{j=0}^\infty b_{j/2}(C)t^{j/2}+\sum_{j=0}^\infty c_{j/2}(C) t^{j/2} \log t$. In the case that the Gaussian curvature $K$ of $(M,g)$ satisfies $|K(p)|\to\infty$ as $p\to C$, we show that $b_{1/2}(C)$ varies irrationally under constant rescalings of the distance circles near the cone point. This is a sharp contrast to the behavior of $b_0(C)$ and of those coefficients $b_j(C)$ which appear in certain known formulas in the case of orbifold cone points or corners of geodesic polygons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22255 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Heat coefficients of surfaces with curved conic singularities Schueth, Dorothee Differential Geometry 58J50 Let $(M,g)$ be a two-dimensional Riemannian manifold of finite diameter with a conical singularity. Under the assumption that the metric near the cone point $C$ is rotationally invariant, but not necessarily flat, we give an explicit formula for the coefficient $b_{1/2}(C)$ in the heat trace expansion $\operatorname{tr}(\operatorname{exp}(-tΔ_g))\sim_{t\searrow0} (4πt)^{-1}\sum_{j=0}^\infty a_j(M) t^j+\sum_{j=0}^\infty b_{j/2}(C)t^{j/2}+\sum_{j=0}^\infty c_{j/2}(C) t^{j/2} \log t$. In the case that the Gaussian curvature $K$ of $(M,g)$ satisfies $|K(p)|\to\infty$ as $p\to C$, we show that $b_{1/2}(C)$ varies irrationally under constant rescalings of the distance circles near the cone point. This is a sharp contrast to the behavior of $b_0(C)$ and of those coefficients $b_j(C)$ which appear in certain known formulas in the case of orbifold cone points or corners of geodesic polygons. |
| title | Heat coefficients of surfaces with curved conic singularities |
| topic | Differential Geometry 58J50 |
| url | https://arxiv.org/abs/2511.22255 |