Constrained Curvature Flows on Pinched Hadamard Surfaces

TL;DR

This paper analyzes area- and length-preserving curvature flows on pinched Hadamard surfaces, establishing long-term behavior, convexity preservation, and convergence properties under various conditions.

Contribution

It extends curvature flow analysis to variable-curvature Hadamard surfaces, providing new results on convexity, long-time existence, and convergence, including a dichotomy in the rotationally symmetric case.

Findings

Convexity is preserved and becomes strict instantly.

Curvature remains bounded and converges under certain conditions.

Flow either converges to a geodesic circle or diverges to infinity, depending on initial conditions.

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.