Journal of Differential Geometry
- J. Differential Geom.
- Volume 105, Number 3 (2017), 427-486.
Boundary torsion and convex caps of locally convex surfaces
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.
Research of the author was supported in part by NSF Grant DMS-1308777, and Simons Collaboration Grant 279374.
J. Differential Geom., Volume 105, Number 3 (2017), 427-486.
Received: 17 February 2015
First available in Project Euclid: 3 March 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Ghomi, Mohammad. Boundary torsion and convex caps of locally convex surfaces. J. Differential Geom. 105 (2017), no. 3, 427--486. doi:10.4310/jdg/1488503004. https://projecteuclid.org/euclid.jdg/1488503004