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March 2017 Boundary torsion and convex caps of locally convex surfaces
Mohammad Ghomi
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J. Differential Geom. 105(3): 427-486 (March 2017). DOI: 10.4310/jdg/1488503004


We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least $4$ times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the $4$ vertex theorem of Sedykh for convex space curves, and thus constitutes a far reaching extension of the classical $4$ vertex theorem. The proof involves studying the arrangement of convex caps in a locally convex surface, and yields a Bose type formula for these objects.

Funding Statement

Research of the author was supported in part by NSF Grant DMS-1308777, and Simons Collaboration Grant 279374.


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Mohammad Ghomi. "Boundary torsion and convex caps of locally convex surfaces." J. Differential Geom. 105 (3) 427 - 486, March 2017.


Received: 17 February 2015; Published: March 2017
First available in Project Euclid: 3 March 2017

zbMATH: 1381.53017
MathSciNet: MR3619309
Digital Object Identifier: 10.4310/jdg/1488503004

Rights: Copyright © 2017 Lehigh University


Vol.105 • No. 3 • March 2017
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