On length-preserving and area-preserving inverse curvature flows in the hyperbolic plane

TL;DR

This paper investigates length- and area-preserving curvature flows of convex curves in hyperbolic space, proving convexity preservation, convergence to geodesic circles, and providing conditions for global existence.

Contribution

It introduces new results on the behavior of inverse curvature flows in hyperbolic geometry, including convergence and existence criteria.

Findings

Convexity is preserved along the flows.

Curves converge smoothly to geodesic circles.

A sufficient condition for global existence is derived.

Abstract

In this paper, we study the area-preserving and length-preserving $\kappa^\alpha$-type curvature flows of smooth, closed, convex curves in the two-dimensional hyperbolic plane $\mathbb H^2$ for $\alpha<0$ and prove that convexity is preserved along the flows. Assuming that the flows exist for all time, we show that the evolving curves converge smoothly to geodesic circles. Furthermore, we also derive a sufficient condition for global existence of the flows.