Hot spots on cones and warped product manifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915713628766208 |
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| author | Hatcher, Lawford |
| author_facet | Hatcher, Lawford |
| contents | We study extrema of solutions to the heat equation (i.e. hot spots) on a class of warped product manifolds of the form $([0,L]\times M,dr^2+f(r)^2h)$ where $(M,h)$ is a closed Riemannian manifold. We prove that, under certain conditions on the warping function $f$, the statement of Rauch's hot spots conjecture holds for the corresponding warped product. We then go on to study the long-time behavior of hot spots on infinite cones over closed Riemannian manifolds. In this case, under appropriate hypotheses on the initial condition, there are four possible long-time behaviors depending only on the spectral gap of the fiber $(M,h)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18054 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hot spots on cones and warped product manifolds Hatcher, Lawford Analysis of PDEs Differential Geometry Spectral Theory 35P05, 35B38, 35J05, 35J25, 58J50 We study extrema of solutions to the heat equation (i.e. hot spots) on a class of warped product manifolds of the form $([0,L]\times M,dr^2+f(r)^2h)$ where $(M,h)$ is a closed Riemannian manifold. We prove that, under certain conditions on the warping function $f$, the statement of Rauch's hot spots conjecture holds for the corresponding warped product. We then go on to study the long-time behavior of hot spots on infinite cones over closed Riemannian manifolds. In this case, under appropriate hypotheses on the initial condition, there are four possible long-time behaviors depending only on the spectral gap of the fiber $(M,h)$. |
| title | Hot spots on cones and warped product manifolds |
| topic | Analysis of PDEs Differential Geometry Spectral Theory 35P05, 35B38, 35J05, 35J25, 58J50 |
| url | https://arxiv.org/abs/2508.18054 |