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Reshetnyak Rigidity for Riemannian Manifolds

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Abstract

We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map \({f:\mathcal{M} \to \mathcal{N}}\) between two oriented Riemannian manifolds, whose differential is almost everywhere an orientation-preserving isometry, is an isometric immersion. This theorem was previously proved using regularity theory for conformal maps; we give a new, simple proof, by generalizing the Piola identity for the cofactor operator. Second, we prove that if there exists a sequence of mapping \({f_n:\mathcal{M} \to \mathcal{N}}\), whose differentials converge in Lp to the set of orientation-preserving isometries, then there exists a subsequence converging to an isometric immersion. These results are generalizations of celebrated rigidity theorems by Liouville (J Math Pures Appl 1850) and Reshetnyak (Sib Mat Zhurnal 8(1):91–114, 1967) from Euclidean to Riemannian settings. Finally, we describe applications of these theorems to non-Euclidean elasticity and to convergence notions of manifolds.

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Acknowledgements

We are grateful to Pavel Giterman,Amitai Yuval andYael Karshon for useful discussions. We also thank Deane Yang for suggesting the current form of Lemma 3.

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Authors and Affiliations

  1. Institute of Mathematics, Hebrew University, Jerusalem, Israel

    Raz Kupferman & Asaf Shachar

  2. Department of Mathematics, University of Toronto, Toronto, Canada

    Cy Maor

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  1. Raz Kupferman
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  2. Cy Maor
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  3. Asaf Shachar
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Correspondence to Asaf Shachar.

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The authors declare that they have no conflict of interest.

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Communicated by D. Kinderlehrer

This research was partially supported by the Israel Science Foundation (Grant No. 1035/17), and by a grant from the Ministry of Science, Technology and Space, Israel and the Russian Foundation for Basic Research, the Russian Federation.

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Kupferman, R., Maor, C. & Shachar, A. Reshetnyak Rigidity for Riemannian Manifolds. Arch Rational Mech Anal 231, 367–408 (2019). https://doi.org/10.1007/s00205-018-1282-9

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