Wilhelm Klingenberg

A proof of the Toponogov Conjecture on complete surfaces

We prove a conjecture Toponogov on complete convex surfaces diffeomorphic to the plane, namely that such surfaces must contain one umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value problem and an existence result for holomorphic discs with Lagrangian boundary conditions, both of which apply to a putative counterexample. This is joint work with Brendan Guilfoyle.