Proof of the Caratheodory Conjecture

TL;DR

This paper proves the Caratheodory conjecture for smooth convex surfaces by linking umbilic points to complex points on Lagrangian surfaces and employing mean curvature flow to construct holomorphic discs.

Contribution

It introduces a novel approach connecting convex surface umbilic points with complex geometry and proves the conjecture using mean curvature flow techniques.

Findings

Proof of the Caratheodory conjecture for $C^{3+ta}$-smooth surfaces.

Establishment of a correspondence between umbilic points and totally real Lagrangian hemispheres.

Construction of holomorphic discs via mean curvature flow with long-time existence.

Abstract

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.