Rigidity of Nonnegatively Curved Surfaces Relative to a Curve

Abstract

Abstract We prove that any properly oriented $\mathcal{C}^{2,1}$ 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 partial differential equations. Our approach also yields a short proof of Cohn-Vossen’s theorem.