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.
"Boundary torsion and convex caps of locally convex surfaces." J. Differential Geom. 105 (3) 427 - 486, March 2017. https://doi.org/10.4310/jdg/1488503004