Rigidity of Nonnegatively Curved Surfaces Relative to a Curve

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 via Hopf's maximum principle.Date: May 15, 2018 (Last Typeset). 2010 Mathematics Subject Classification. Primary: 53A05, 35J96; Secondary 58J30, 52A15.

Nonrigidity of flat ribbons

“…[9]. This is in striking contrast to the case of positive Gaussian curvature, where a surface is globally rigid relative to any of its curves [10].…”