Stability of isometric immersions of hypersurfaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866911822178680832 |
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| author | Alpern, Itai Kupferman, Raz Maor, Cy |
| author_facet | Alpern, Itai Kupferman, Raz Maor, Cy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_06654 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stability of isometric immersions of hypersurfaces Alpern, Itai Kupferman, Raz Maor, Cy Differential Geometry Analysis of PDEs 53C24, 53C42, 74B20, 74K25 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. |
| title | Stability of isometric immersions of hypersurfaces |
| topic | Differential Geometry Analysis of PDEs 53C24, 53C42, 74B20, 74K25 |
| url | https://arxiv.org/abs/2306.06654 |