Finding umbilics on open convex surfaces

@article{Fontenele2019FindingUO,
  title={Finding umbilics on open convex surfaces},
  author={Francisco Fontenele and Frederico Xavier},
  journal={Revista Matem{\'a}tica Iberoamericana},
  year={2019}
}
By the Poincare–Hopf theorem, every ovaloid has at least one umbilic. In this paper we extend this result to the more general case of complete positively curved surfaces in R3 whose shape operator A satisfies inf |A|>0 and sup |∇A|<∞. 
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Proof of the Toponogov Conjecture on Complete Surfaces

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…

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