Georgia Institute of Technology College of Sciences

We prove that the torsion of any smooth closed curve in Euclidean space which bounds a simply connected locally convex surface vanishes at least 4 times (vanishing of torsion means that the first 3 derivatives of the curve are linearly dependent). This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in 3-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 for planar curves. The proof follows from an extensive study of the structure of convex caps in a locally convex surface.