Rigidity of nonnegatively curved surfaces relative to a curve

TL;DR

This paper proves a local rigidity theorem for nonnegatively curved surfaces in Euclidean space, showing they are uniquely determined by their boundary values, extending classical results with new conditions and methods.

Contribution

It extends Cohn-Vossen's rigidity theorem to nonnegatively curved surfaces with boundary conditions, using elliptic PDE unique continuation principles.

Findings

Unique determination of positively curved surfaces by boundary data

Extension of rigidity to nonnegatively curved surfaces under certain conditions

Short proof of Cohn-Vossen's rigidity theorem

Abstract

We prove that any properly oriented $C^{2,1}$ isometric immersion of a positively curved Riemannian surface M into Euclidean 3-space is uniquely determined, up to a rigid motion, by its values on any curve segment in M. A generalization of this result to nonnegatively curved surfaces is presented as well under suitable conditions on their parabolic points. Thus we obtain a local version of Cohn-Vossen's rigidity theorem for convex surfaces subject to a Dirichlet condition. The proof employs in part Hormander's unique continuation principle for elliptic PDEs. Our approach also yields a short proof of Cohn-Vossen's theorem.