Proof of the Toponogov Conjecture on Complete Surfaces

Authors

Brendan Guilfoyle, School of STEM, Munster Technological University, Kerry, Tralee Co., Kerry, IrelandFollow
Wilhelm Klingenberg, Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, UK

ORCID

https://orcid.org/0000-0002-7437-5046

Document Type

Preprint

Disciplines

Geometry and Topology | Mathematics | Physical Sciences and Mathematics

Publication Details

Article current available on arXiv. Soon to be published by Journal of Gökova Geometry Topology, vol. 17 (2024).

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.

Recommended Citation

Guilfoyle, Brendan and Klingenberg, Wilhelm, "Proof of the Toponogov Conjecture on Complete Surfaces" (2024). Department of Mathematics Publications [online].
Available at: https://doi.org/10.48550/arXiv.2002.12787