Revista Matemática Iberoamericana

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Volume 35, Issue 7, 2019, pp. 2035–2052
DOI: 10.4171/rmi/1109

Published online: 2019-07-17

Finding umbilics on open convex surfaces

Francisco Fontenele[1] and Frederico Xavier[2]

(1) Universidade Federal Fluminense, Niterói, Brazil
(2) Texas Christian University, Fort Worth, USA

By the Poincaré–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 $\mathbb R^3$ whose shape operator $A$ satisfies inf $|A|>0$ and sup $|\nabla A|<\infty$.

Keywords: Existence of umbilics, Milnor conjecture, lines of curvature, index of umbilics

Fontenele Francisco, Xavier Frederico: Finding umbilics on open convex surfaces. Rev. Mat. Iberoam. 35 (2019), 2035-2052. doi: 10.4171/rmi/1109