Curve shortening flows in warped product manifolds

TL;DR

This paper investigates the behavior of curve shortening flows in specific warped product manifolds, demonstrating long-term existence and convergence to geodesic curves for initial graphs.

Contribution

It establishes the global existence and convergence of curve shortening flows in warped product manifolds with particular metrics, extending understanding of geometric flows in these settings.

Findings

Flow exists for all time for initial graphs.

Flow converges to a geodesic closed curve.

Results apply to manifolds with warped metrics on $S^1 \times N$.

Abstract

We study curve shortening flows in two types of warped product manifolds. These manifolds are $S^1\times N$ with two types of warped metrics where $S^1$ is the unit circle in $R^2$ and $N$ is a closed Riemannian manifold. If the initial curve is a graph over $S^1$, then its curve shortening flow exists for all times and finally converges to a geodesic closed curve.