Boundary torsion and convex caps of locally convex surfaces. (English) Zbl 1381.53017

Ghomi, Mohammad

J. Differ. Geom. 105, No. 3, 427-487 (2017).

The author investigates an open question by Yau, regarding when a closed curve in three-space bounds a disk of positive curvature. Following up on prior conjecture, he finds that the curve must have at least four vertices, establishing a new requirement for the existence of such curves. The majority of the paper is dedicated to proving this result. However, the last section contains a collection of examples to cast some relief on the proof, showing applications to real curves.

Reviewer: James P. Howard II (Columbia)

MSC:

53A05	Surfaces in Euclidean and related spaces
53A04	Curves in Euclidean and related spaces
57R40	Embeddings in differential topology

Keywords:

convex caps; closed space curves; Bose’s formula; topological disks; vertex