  A well-known conjecture of Caratheodory states that the number of umbilic
points on a closed convex surface in ${\mathbb E}^3$ must be greater than one.
In this paper we prove this for $C^{3+\alpha}$-smooth surfaces. The Conjecture
is first reformulated in terms of complex points on a Lagrangian surface in
$TS^2$, viewed as the space of oriented geodesics in ${\mathbb E}^3$. Here
complex and Lagrangian refer to the canonical neutral Kaehler structure on
$TS^2$. We then prove that the existence of a closed convex surface with only
one umbilic point implies the existence of a totally real Lagrangian hemisphere
in $TS^2$, to which it is not possible to attach the edge of a holomorphic
disc. The main step in the proof is to establish the existence of a holomorphic
disc with edge contained on any given totally real Lagrangian hemisphere. To
construct the holomorphic disc we utilize mean curvature flow with respect to
the neutral metric. Long-time existence of this flow is proven by a priori
estimates and we show that the flowing disc is asymptotically holomorphic.
Existence of a holomorphic disc is then deduced from Schauder estimates.
