Stability of isometric immersions of hypersurfaces
| Autor: | Alpern, Itai, Kupferman, Raz, Maor, Cy |
|---|---|
| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Forum of Mathematics, Sigma. 2024;12:e43 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1017/fms.2024.30 |
| Popis: | 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. Comment: vr2: some references added. vr3: improvements in presentation, Section 2.1 added, Appendix B removed |
| Databáze: | arXiv |
| Externí odkaz: |
|