Boundary torsion and convex caps of locally convex surfaces

TL;DR

This paper proves that the torsion of boundary curves of locally convex surfaces in space vanishes at least four times, extending classical theorems and answering longstanding questions in differential geometry.

Contribution

It establishes a lower bound on torsion zeros for boundary curves of locally convex surfaces, generalizing the 4 vertex theorem to a broader class of surfaces.

Findings

Torsion of boundary curves vanishes at least 4 times

Generalization of Sedykh's 4 vertex theorem

A Bose type formula for convex caps

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.