Reshetnyak Rigidity for Riemannian Manifolds

Raz Kupferman, Cy Maor, Asaf Shachar*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map f: M→ 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 fn: M→ 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.

Original languageAmerican English
Pages (from-to)367-408
Number of pages42
JournalArchive for Rational Mechanics and Analysis
Volume231
Issue number1
DOIs
StatePublished - 22 Jan 2019

Bibliographical note

Funding Information:
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.

Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

Fingerprint

Dive into the research topics of 'Reshetnyak Rigidity for Riemannian Manifolds'. Together they form a unique fingerprint.

Cite this