Wilhelm Klingenberg (University of Durham)
wilhelm.klingenberg@durham.ac.uk
A well-known conjecture of Carathéodory states that the number of umbilic points on a closed convex surface in $\mathbb{R}^3$ must be greater than one. In our talk we will outline our proof which was achieved in collaboration with Brendan Guilfoyle.
The Conjecture is first reformulated in terms of complex points on a Lagrangian surface in $TS^2$, viewed as the space of oriented lines in $\mathbb{R}^3$. 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 boundary of a holomorphic disc. The main step in the proof is to establish the existence of a holomorphic disc with boundary contained on any given totally real Lagrangian hemisphere. To construct the holomorphic disc we use a capillary-type boundary value problem for 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 a version of compactness for $J$-holomorphic curves with boundary contained in a totally real surface.