Finding umbilics on open convex surfaces

Fontenele, Francisco; Xavier, Frederico

doi:10.4171/rmi/1109

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Introduction and Results1

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2020

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“…The final corollary comes from Properties I and III excluding Property II, which establishes the convex case of a conjecture attributed to John Milnor from 1965 [7] -see [1] for a recent discussion. Corollary 6.…”

Section: Introduction and Resultsmentioning

confidence: 74%

Proof of the Toponogov Conjecture on Complete Surfaces

Guilfoyle,

Klingenberg

2020

Preprint

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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 singularitytype theorem, as well as the proof of a conjecture of Milnor's from 1965 in the convex case.

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Section: Introduction and Resultsmentioning

confidence: 74%