TY - JOUR

T1 - Reshetnyak Rigidity for Riemannian Manifolds

AU - Kupferman, Raz

AU - Maor, Cy

AU - Shachar, Asaf

N1 - Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/1/22

Y1 - 2019/1/22

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85050269937&partnerID=8YFLogxK

U2 - 10.1007/s00205-018-1282-9

DO - 10.1007/s00205-018-1282-9

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85050269937

SN - 0003-9527

VL - 231

SP - 367

EP - 408

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

IS - 1

ER -