  We prove a conjecture of Toponogov on complete convex surfaces, namely that
such surfaces must contain an 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.
  Corollaries of the main theorem include a Hawking-Penrose singularity-type
theorem, as well as the proof of a conjecture of Milnor's from 1965 in the
convex case.
