The evolution of Jordan curves on S 2 by curve shortening flow
Abstract
In this paper we prove that if
γ
is a Jordan curve on
S
2
then there is a smooth curve shortening flow defined on
(0,T)
which converges to
γ
in
C
0
as
t→
0
+
. Another perspective is that the level-set flow of
γ
is smooth. This is a generalization of the author's previous work where the planar case was studied. If a Jordan curve on
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
S
2
.