Proof of the Toponogov Conjecture on Complete Surfaces

Abstract

We prove a conjecture of Toponogov on complete convex planes, namely that such planes 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.

AMS Latex, 44 pages, 3 figures. The paper includes previously unpublished material by the authors that originally appeared in arXiv:0808.0851

Cited by in corpus (2)

• On Isolated Umbilic Points
• From CT scans to 4-manifold topology