Hot spots in domains of constant curvature
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915450312458240 |
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| author | Hatcher, Lawford |
| author_facet | Hatcher, Lawford |
| contents | We prove constant-curvature analogues of several results regarding the hot spots conjecture in dimension two. Our main theorem shows that the hot spots conjecture holds for all non-acute geodesic triangles of constant negative curvature. We also prove that, under certain circumstances, on constant (positive or negative) curvature triangles, first mixed Dirichlet-Neumann Laplace eigenfunctions have no non-vertex critical points. Moreover, we show that each of these eigenfunctions is monotonic with respect to some Killing field. Finally, we show that for general simply connected polygons of non-zero constant curvature--with exactly one family of exceptions--second Neumann eigenfunctions of the Laplacian have at most finitely many critical points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13353 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hot spots in domains of constant curvature Hatcher, Lawford Spectral Theory Analysis of PDEs Differential Geometry 35P05, 35B38, 35J05, 35J25, 58J50 We prove constant-curvature analogues of several results regarding the hot spots conjecture in dimension two. Our main theorem shows that the hot spots conjecture holds for all non-acute geodesic triangles of constant negative curvature. We also prove that, under certain circumstances, on constant (positive or negative) curvature triangles, first mixed Dirichlet-Neumann Laplace eigenfunctions have no non-vertex critical points. Moreover, we show that each of these eigenfunctions is monotonic with respect to some Killing field. Finally, we show that for general simply connected polygons of non-zero constant curvature--with exactly one family of exceptions--second Neumann eigenfunctions of the Laplacian have at most finitely many critical points. |
| title | Hot spots in domains of constant curvature |
| topic | Spectral Theory Analysis of PDEs Differential Geometry 35P05, 35B38, 35J05, 35J25, 58J50 |
| url | https://arxiv.org/abs/2508.13353 |