  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 $\mathbb{S}^2$.

Analysis of PDEs;Differential Geometry;General Topology