Constrained Curvature Flows on Pinched Hadamard Surfaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914475478614016 |
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| author | Albert-Niclòs, Sara Cabezas-Rivas, Esther |
| author_facet | Albert-Niclòs, Sara Cabezas-Rivas, Esther |
| contents | We study area- and length-preserving curvature flows for embedded closed curves on pinched Hadamard surfaces. In the variable-curvature setting, the evolution equations contain additional lower-order terms, so the PDE analysis requires refined comparison arguments and delicate curvature estimates. For smooth convex initial curves, we prove preservation and instantaneous strictness of convexity, long-time existence, and uniform bounds for the curvature and its higher derivatives. Under additional geometric assumptions, we obtain convergence of the curvature to a constant. In the rotationally symmetric case, the area-preserving flow exhibits a dichotomy: either the evolving curves converge exponentially to a geodesic circle, or they drift off to infinity and approach a constant-curvature limit curve. We also identify a geometric condition on the initial curve that prevents escape to infinity and guarantees convergence to a geodesic circle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13734 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Constrained Curvature Flows on Pinched Hadamard Surfaces Albert-Niclòs, Sara Cabezas-Rivas, Esther Differential Geometry Analysis of PDEs 53E10, 35R01, 35K59, 35K93, 35B35 We study area- and length-preserving curvature flows for embedded closed curves on pinched Hadamard surfaces. In the variable-curvature setting, the evolution equations contain additional lower-order terms, so the PDE analysis requires refined comparison arguments and delicate curvature estimates. For smooth convex initial curves, we prove preservation and instantaneous strictness of convexity, long-time existence, and uniform bounds for the curvature and its higher derivatives. Under additional geometric assumptions, we obtain convergence of the curvature to a constant. In the rotationally symmetric case, the area-preserving flow exhibits a dichotomy: either the evolving curves converge exponentially to a geodesic circle, or they drift off to infinity and approach a constant-curvature limit curve. We also identify a geometric condition on the initial curve that prevents escape to infinity and guarantees convergence to a geodesic circle. |
| title | Constrained Curvature Flows on Pinched Hadamard Surfaces |
| topic | Differential Geometry Analysis of PDEs 53E10, 35R01, 35K59, 35K93, 35B35 |
| url | https://arxiv.org/abs/2604.13734 |