Wilhelm Klingenberg (Durhham University) Proof of the Toponogov Conjecture on Complete Surfaces
Abstract: In 1995, Victor Andreevich Toponogov [1] authored the following conjecture: “Every smooth non-compact strictly convex and complete surface of genus zero has an umbilic point, possibly at infinity“. In our talk, we will outline the 2024 proof in collaboration with Brendan Guilfoyle [2]. Namely we prove that, should a counter-example to the Conjecture exist, (a) the Fredholm index of an associated Riemann Hilbert boundary problem for holomorphic discs is negative [3]. Thereby, (b) no such holomorphic discs survive for a generic perturbation of the boundary condition (these form a Banach manifold under the assumption that the Conjecture is incorrect). Finally, however, (c) the geometrization by a neutral Kaehler metric [4] of the associated model allows for Mean Curvature Flow [5] with mixed Dirichlet – Neumann boundary conditions to generate a holomorphic disc from an initial spacelike disc. This completes the indirect proof of said conjecture as (b) and (c) are in contradiction.
More details on http://geometry.ma.ic.ac.uk/seminar/