The evolution of Jordan curves on $\mathbb{S}^2$ by curve shortening flow

TL;DR

This paper proves that Jordan curves on the sphere can be smoothly approximated by curve shortening flow, with the flow's behavior depending on the measure of the initial curve, extending previous planar results to the sphere.

Contribution

The work generalizes curve shortening flow results from the plane to the sphere, establishing existence, convergence, and measure-dependent behavior of the flow for Jordan curves on $\

Findings

Flow converges to the initial curve in $\

Level-set flow becomes smooth or forms an annulus depending on measure

Length estimates depend on the $r$-multiplicity, extended to $\

Abstract

In this paper we prove that if $\gamma$ is a Jordan curve on $\mathbb{S}^2$ then there is a smooth curve shortening flow defined on $(0,T)$ which converges to $\gamma$ in $\mathcal{C}^0$ as $t\to 0^+ $. Another perspective is that the level-set flow of $\gamma$ is smooth. This is a generalization of the author's previous work where the planar case was studied. If a Jordan curve on $\mathbb{S}^2$ has Lebesgue measure zero then we show that the level-set flow instantly becomes a smooth closed curve. If the Lebesgue measure is positive then for small time the level-set flow is an annulus with smooth boundary. This second case should be interpreted as a failure of uniqueness. As in the planar case a key step in the proof is establishing a length estimate for smooth curves that depends on a geometric quantity called the $r$-multiplicity. The majority of this paper concerns the extension of this length estimate to $\mathbb{S}^2$.