Stability of isometric immersions of hypersurfaces

Abstract

We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to $L^p$-perturbations of their fundamental forms: For a manifold $M^d$ endowed with a reference metric and a reference shape operator, we show that a sequence of immersions $f_n:M^d\to N^{d+1}$, whose pullback metrics and shape operators are arbitrary close in $L^p$ 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 by the authors to a general target manifold $N$, removing a constant curvature assumption. The method of proof differs from that in Alpern et al.: 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 Ciarlet et al. (Anal. Appl. 2019) but with no a-priori assumed bounds.

arXiv commentvr2: some references added. vr3: improvements in presentation, Section 2.1 added, Appendix B removed

References in corpus (8)

• Elastic theory of unconstrained non-Euclidean plates
• A Riemannian Approach to Reduced Plate, Shell, and Rod Theories
• Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics
• On the role of curvature in the elastic energy of non-Euclidean thin bodies
• Reshetnyak rigidity for Riemannian manifolds
• On Asymptotic Rigidity and Continuity Problems in Nonlinear Elasticity on Manifolds and Hypersurfaces
• Asymptotic rigidity for shells in non-Euclidean elasticity
• Rigidity and regularity of co-dimension one Sobolev isometric immersions

Cited by in corpus (1)

• Linearization in incompatible elasticity for general ambient spaces