Rigidity of nonnegatively curved surfaces relative to a curve

Mohammad Ghomi; Joel Spruck

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arXiv · 1805.02580

Rigidity of nonnegatively curved surfaces relative to a curve

Mohammad Ghomi·Joel Spruck

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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.

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Mohammad Ghomi, Joel Spruck. 2018-05-07. Rigidity of nonnegatively curved surfaces relative to a curve. https://arxiv.org/abs/1805.02580

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