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

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 $\mathbb{S}^2$.