As on arXiv
Abstract
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.
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Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere
Topics Long-time existence & convergence, Other curvature flows · Same authors Sara Albert-Niclòs, Esther Cabezas-Rivas · Shared terms “preserving curvature”, “preservation”
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On a non-local area-preserving curve flow
Topics Volume-preserving & constrained flows, Other curvature flows · Shared terms “area preserving”, “curves converge”, “evolving curves”
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On length-preserving and area-preserving inverse curvature flows in the hyperbolic plane
Topics Volume-preserving & constrained flows, Long-time existence & convergence, Other curvature flows · Shared terms “length preserving”, “geodesic circle”, “area preserving”
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Geometric -stability of geodesic spheres under volume-preserving mean curvature flow
Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same author Esther Cabezas-Rivas · Shared terms “curvature estimates”
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Asymptotic circularity of immortal area-preserving curvature flows
Topics Volume-preserving & constrained flows, Rotational symmetry & warped products, Long-time existence & convergence · Shared terms “area preserving”
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The weighted geometric inequalities for static convex domains in static rotationally symmetric spaces
Topics Volume-preserving & constrained flows, Rotational symmetry & warped products, Long-time existence & convergence · Shared terms “rotationally symmetric”
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The quermassintegral preserving mean curvature flow in the sphere
Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same author Esther Cabezas-Rivas · Shared terms “constant curvature”, “convex initial”
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Area-preserving curvature flow for open curves with general contact angles on skew lines
Topics Volume-preserving & constrained flows, Long-time existence & convergence, Other curvature flows · Shared terms “area preserving”, “initial curves”, “evolving curves”
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