Relative isometric embeddings of Riemannian manifolds

Ghomi, Mohammad; Greene, Robert E.

doi:10.1090/s0002-9947-2010-05095-0

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Cited by 2 publications

References 18 publications

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Rigidity of Nonnegatively Curved Surfaces Relative to a Curve

Ghomi

Spruck

2018

International Mathematics Research Notices

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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.

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Rigidity of Nonnegatively Curved Surfaces Relative to a Curve

Ghomi

Spruck

2018

International Mathematics Research Notices

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Global Nash-Kuiper theorem for compact manifolds

Cao¹,

Székelyhidi²

2019

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We obtain global extensions of the celebrated Nash-Kuiper theorem for C 1,θ isometric immersions of compact manifolds with optimal Hölder exponent. In particular for the Weyl problem of isometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity prevails upto exponent θ < 1/5. This extends previous results on embedding 2-discs as well as higher dimensional analogues.

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Relative isometric embeddings of Riemannian manifolds

Abstract: Abstract. We prove the existence of C 1 isometric embeddings, and C ∞ approximate isometric embeddings, of Riemannian manifolds into Euclidean space with prescribed values in a neighborhood of a point.

Cited by 2 publications

References 18 publications

Rigidity of Nonnegatively Curved Surfaces Relative to a Curve

Rigidity of Nonnegatively Curved Surfaces Relative to a Curve

Global Nash-Kuiper theorem for compact manifolds

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