Proof of the Caratheodory Conjecture

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 $C^{3+\hat{I}\pm }$-smooth surfaces. The Conjecture is first reformulated in terms of complex points on a Lagrangian surface in $T{S}^2$, viewed as the space of oriented geodesics in ${\mathbb{E}}^3$. Here complex and Lagrangian refer to the canonical neutral Kaehler structure on $T{S}^2$. 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 $T{S}^2$, 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.

This work has appeared in three parts in the references below: Fredholm regularity of Section 2 in the first reference, higher codimension mean curvature flow of Section 3 in the second reference and proof of existence of a holomorphic disc of Sections 4, 5 and 6 in the third. The three individual papers may be found at arxiv:1812.00707, arxiv:1812.00710 and arxiv:2002.12787 respectively