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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 convergesto $\gamma$ in $\mathcal{C}^0$ as $t\to 0^+ $. Another perspective is that thelevel-set flow of $\gamma$ is smooth. This is a generalization of the author'sprevious 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 flowinstantly becomes a smooth closed curve. If the Lebesgue measure is positivethen for small time the level-set flow is an annulus with smooth boundary. Thissecond case should be interpreted as a failure of uniqueness.  As in the planar case a key step in the proof is establishing a lengthestimate for smooth curves that depends on a geometric quantity called the$r$-multiplicity. The majority of this paper concerns the extension of thislength estimate to $\mathbb{S}^2$.

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