Stability of isometric immersions of hypersurfaces
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 ${\mathcal M}^d$ endowed with a reference metric and a reference shape operator, we show that a sequence of immer...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000306/type/journal_article |
| _version_ | 1797229716972765184 |
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| author | Itai Alpern Raz Kupferman Cy Maor |
| author_facet | Itai Alpern Raz Kupferman Cy Maor |
| author_sort | Itai Alpern |
| collection | DOAJ |
| description | 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
${\mathcal M}^d$
endowed with a reference metric and a reference shape operator, we show that a sequence of immersions
$f_n:{\mathcal M}^d\to {\mathcal N}^{d+1}$
, 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 [AKM22] to a general target manifold
${\mathcal N}$
, 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. |
| first_indexed | 2024-04-24T15:17:01Z |
| format | Article |
| id | doaj.art-57195083cec542e4a470b2ae655ab0e1 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-24T15:17:01Z |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-57195083cec542e4a470b2ae655ab0e12024-04-02T09:10:58ZengCambridge University PressForum of Mathematics, Sigma2050-50942024-01-011210.1017/fms.2024.30Stability of isometric immersions of hypersurfacesItai Alpern0Raz Kupferman1https://orcid.org/0000-0001-7699-7466Cy Maor2https://orcid.org/0000-0002-7948-0945Einstein Institute of Mathematics, The Hebrew University, Jerusalem, Israel; E-mail: .Einstein Institute of Mathematics, The Hebrew University, Jerusalem, IsraelEinstein Institute of Mathematics, The Hebrew University, Jerusalem, Israel; E-mail: .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 ${\mathcal M}^d$ endowed with a reference metric and a reference shape operator, we show that a sequence of immersions $f_n:{\mathcal M}^d\to {\mathcal N}^{d+1}$ , 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 [AKM22] to a general target manifold ${\mathcal N}$ , 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.https://www.cambridge.org/core/product/identifier/S2050509424000306/type/journal_article53C2453C4274B2074K25 |
| spellingShingle | Itai Alpern Raz Kupferman Cy Maor Stability of isometric immersions of hypersurfaces Forum of Mathematics, Sigma 53C24 53C42 74B20 74K25 |
| title | Stability of isometric immersions of hypersurfaces |
| title_full | Stability of isometric immersions of hypersurfaces |
| title_fullStr | Stability of isometric immersions of hypersurfaces |
| title_full_unstemmed | Stability of isometric immersions of hypersurfaces |
| title_short | Stability of isometric immersions of hypersurfaces |
| title_sort | stability of isometric immersions of hypersurfaces |
| topic | 53C24 53C42 74B20 74K25 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000306/type/journal_article |
| work_keys_str_mv | AT itaialpern stabilityofisometricimmersionsofhypersurfaces AT razkupferman stabilityofisometricimmersionsofhypersurfaces AT cymaor stabilityofisometricimmersionsofhypersurfaces |