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Nonlinear estimates for hypersurfaces in terms of their fundamental forms.

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Title: Nonlinear estimates for hypersurfaces in terms of their fundamental forms.
Authors: Malin, Maria1 malinmaria@yahoo.com, Mardare, Cristinel2 mardare@ann.jussieu.fr
Source: Comptes Rendus. Mathématique. Nov2017, Vol. 355 Issue 11, p1196-1200. 5p.
Subjects: Hypersurfaces, Immersions (Mathematics), Euclidean geometry, Vectors (Calculus), Matrices (Mathematics), Differentiable manifolds
Abstract (English): A sufficiently regular hypersurface immersed in the ( n + 1 ) -dimensional Euclidean space is determined up to a proper isometry of R n + 1 by its first and second fundamental forms. As a consequence, a sufficiently regular hypersurface with boundary, whose position and positively-oriented unit normal vectors are given on a non-empty portion of its boundary, is uniquely determined by its first and second fundamental forms. We establish here stronger versions of these uniqueness results by means of inequalities showing that an appropriate distance between two immersions from a domain ω of R n into R n + 1 is bounded by the L p -norm of the difference between matrix fields defined in terms of the first and second fundamental forms associated with each immersion. [ABSTRACT FROM AUTHOR]
Abstract (French): Résumé Une hypersuface suffisamment régulière immergée dans l'espace euclidien ( n + 1 ) -dimensionnel est determinée à une isométrie propre de R n + 1 près par ses deux premières formes fondamentales. Par conséquent, une hypersuface suffisamment régulière avec frontière, dont la position et les vecteurs unitaires normaux et orientés positivement sont donnés sur une partie non vide de sa frontière, est déterminée uniquement par ses deux premières formes fondamentales. Nous établissons ici des versions plus fortes de ce résultat, en établissant des inégalités montrant qu'une distance appropriée entre deux immersions d'un domaine ω de R n dans R n + 1 est majorée par la norme L p de la difference entre des champs de matrices définies en fonction des deux premières formes fondamentales associées à chaque immersion. [ABSTRACT FROM AUTHOR]
Copyright of Comptes Rendus. Mathématique is the property of Academie des Science, Centre Mersenne and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Nonlinear estimates for hypersurfaces in terms of their fundamental forms.
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  Data: <searchLink fieldCode="AR" term="%22Malin%2C+Maria%22">Malin, Maria</searchLink><relatesTo>1</relatesTo><i> malinmaria@yahoo.com</i><br /><searchLink fieldCode="AR" term="%22Mardare%2C+Cristinel%22">Mardare, Cristinel</searchLink><relatesTo>2</relatesTo><i> mardare@ann.jussieu.fr</i>
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  Data: <searchLink fieldCode="JN" term="%22Comptes+Rendus%2E+Mathématique%22">Comptes Rendus. Mathématique</searchLink>. Nov2017, Vol. 355 Issue 11, p1196-1200. 5p.
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  Data: <searchLink fieldCode="DE" term="%22Hypersurfaces%22">Hypersurfaces</searchLink><br /><searchLink fieldCode="DE" term="%22Immersions+%28Mathematics%29%22">Immersions (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Euclidean+geometry%22">Euclidean geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Vectors+%28Calculus%29%22">Vectors (Calculus)</searchLink><br /><searchLink fieldCode="DE" term="%22Matrices+%28Mathematics%29%22">Matrices (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Differentiable+manifolds%22">Differentiable manifolds</searchLink>
– Name: Abstract
  Label: Abstract (English)
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  Data: A sufficiently regular hypersurface immersed in the ( n + 1 ) -dimensional Euclidean space is determined up to a proper isometry of R n + 1 by its first and second fundamental forms. As a consequence, a sufficiently regular hypersurface with boundary, whose position and positively-oriented unit normal vectors are given on a non-empty portion of its boundary, is uniquely determined by its first and second fundamental forms. We establish here stronger versions of these uniqueness results by means of inequalities showing that an appropriate distance between two immersions from a domain ω of R n into R n + 1 is bounded by the L p -norm of the difference between matrix fields defined in terms of the first and second fundamental forms associated with each immersion. [ABSTRACT FROM AUTHOR]
– Name: Abstract
  Label: Abstract (French)
  Group: Ab
  Data: Résumé Une hypersuface suffisamment régulière immergée dans l'espace euclidien ( n + 1 ) -dimensionnel est determinée à une isométrie propre de R n + 1 près par ses deux premières formes fondamentales. Par conséquent, une hypersuface suffisamment régulière avec frontière, dont la position et les vecteurs unitaires normaux et orientés positivement sont donnés sur une partie non vide de sa frontière, est déterminée uniquement par ses deux premières formes fondamentales. Nous établissons ici des versions plus fortes de ce résultat, en établissant des inégalités montrant qu'une distance appropriée entre deux immersions d'un domaine ω de R n dans R n + 1 est majorée par la norme L p de la difference entre des champs de matrices définies en fonction des deux premières formes fondamentales associées à chaque immersion. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Comptes Rendus. Mathématique is the property of Academie des Science, Centre Mersenne and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.crma.2017.10.014
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        Text: English
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        PageCount: 5
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        Type: general
      – SubjectFull: Immersions (Mathematics)
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      – SubjectFull: Euclidean geometry
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      – SubjectFull: Matrices (Mathematics)
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      – SubjectFull: Differentiable manifolds
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      – TitleFull: Nonlinear estimates for hypersurfaces in terms of their fundamental forms.
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              Text: Nov2017
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