Wilhelm Klingenberg (Durham). Proof of the Toponogov Conjecture on Complete Surfaces Friday 2nd May, 1:30-2:30pm. Huxley 140.
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.
References :
[1] V.A. Toponogov, (1995) On conditions for existence of umbilical points on a convex surface, Siberian Mathematical Journal, 36 780–784.
[2] B. Guilfoyle and W. Klingenberg (2024) Proof of the Toponogov Conjecture on complete surfaces, J. Gokova Geom. Topol. GGT 17 1–50.
[3] Guilfoyle, B., & Klingenberg, W. (2020) Fredholm-regularity of holomorphic discs in plane bundles over compact surfaces. Annales de la Faculté des sciences de Toulouse (En ligne), 29(3), 565-576.
[4] B. Guilfoyle and W. Klingenberg (2005) An indefinite Kaehler metric on the space of oriented lines, J. London Math. Soc. 72.2, 497–509.
[5] B. Guilfoyle and W. Klingenberg, Higher codimensional mean curvature flow of compact spacelike submanifolds, Trans. Amer. Math. Soc. 372.9 (2019) 6263–6281.
Masafumi Hattori (Nottingham). TBA Friday 9th May, 1:30-2:30pm. Huxley 140.
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Shaked Bader (Oxford). TBA Friday 16th May, 1:30-2:30pm. Huxley 140.
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Matt Booth (Imperial). TBA Friday 23rd May, 1:30-2:30pm. Huxley 140.
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Yuhan Sun (Imperial). TBA Friday 30th May, 1:30-2:30pm. Huxley 140.
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Matthew Hedden (Michigan State University). TBA Friday 6th June, 1:30-2:30pm. Huxley 140.
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Anya Nordskova (University of Hasselt). TBA Friday 13th June, 1:30-2:30pm. Huxley 140.
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Daniel Bath (KU Leuven). TBA Friday 20th June, 1:30-2:30pm. Huxley 140.
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Silvia Sabatini (University of Cologne). TBA Friday 27th June, 1:30-2:30pm. Huxley 140.
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