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.
- Pub Date:
- January 2015
- Mathematics - Differential Geometry;
- Mathematics - Analysis of PDEs;
- Mathematics - Geometric Topology;
- Mathematics - Metric Geometry;
- 53 pages, 23 figures