Boundary torsion and convex caps of locally convex surfaces
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 January 2015
 arXiv:
 arXiv:1501.07626
 Bibcode:
 2015arXiv150107626G
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs;
 Mathematics  Geometric Topology;
 Mathematics  Metric Geometry;
 53A04;
 53A07;
 53C23;
 53C45
 EPrint:
 53 pages, 23 figures