Stability of isometric immersions of hypersurfaces

Itai Alpern, Raz Kupferman*, Cy Maor

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to Lp-perturbations of their fundamental forms: For a manifold Md endowed with a reference metric and a reference shape operator, we show that a sequence of immersions fn : Md → Nd+1, whose pullback metrics and shape operators are arbitrary close in Lp to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result [AKM22] to a general target manifold N, removing a constant curvature assumption. The method of proof differs from that in [AKM22]: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to [CMM19] but with no a priori assumed bounds.

Original languageAmerican English
Article numbere43
JournalForum of Mathematics, Sigma
Volume12
DOIs
StatePublished - 2 Apr 2024

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Publisher Copyright:
© The Author(s), 2024.

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