The Dirichlet problem for curve shortening flow

TL;DR

This paper studies how open curves with fixed endpoints evolve under curve shortening flow in convex domains on surfaces of constant curvature, showing they become geodesics without forming singularities.

Contribution

It extends the understanding of curve shortening flow to open curves with fixed endpoints in curved surfaces, establishing long-term behavior and convergence to geodesics.

Findings

Open curves do not develop singularities under the flow.

Curves evolve to geodesics in convex domains.

The analysis uses Huisken's distance comparison technique.

Abstract

We investigate the evolution of open curves with fixed endpoints under the curve shortening flow, which evolves curves in proportion to their curvature. Using a distance comparison of Huisken, we determine the long-term behavior of open curves with fixed endpoints evolving in certain convex domains on surfaces of constant curvature. Specifically, we show that such curves do not develop singularities, and evolve to geodesics.