Abstract

We outline joint work with Brendan Guilfoyle, which establishes a proof of the Caratheodory Conjecture. This claims that every C3 - differentiable sphere in Euclidean space admits at least two umbilic points. (These are locally spherical points; at such points both principal curvatures are equal, and every tangent vector is a principal direction). Remark: This is one more umbilic than needs to appear for topological reasons, namely the nonvanishing of the Euler number of the sphere (thereby presents an instance of rigidity). Our proof is inspired by Gromov’s symplectic rigidity-flexibility dichotomy (specifically by his approach to the rigidity of convex surfaces which lead him to the development of his theory of pseudoholomorphic curves). It uses new a – priory gradient estimates for Mean Curvature Flow in manifolds of split signature (building on work of Bartnik and Ecker-Huisken). The latter allows us to construct a holomorphic disc with boundary encircling an isolated umbilic point (in a symplectic model space). This results in sufficient rigidity to prove CC in the spirit of said dichotomy.