New compatibility conditions for the fundamental theorem of surface theory

Philippe G. Ciarlet; Liliana Gratie; Cristinel Mardare

doi:10.1016/j.crma.2007.07.014

Differential Geometry

New compatibility conditions for the fundamental theorem of surface theory

Presented by: Philippe G. Ciarlet

Philippe G. Ciarlet ¹; Liliana Gratie ²; Cristinel Mardare ³

¹ Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
² Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
³ Laboratoire Jacques-Louis Lions, Université Pierre-et-Marie-Curie, 4, place Jussieu, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 273-278.

Abstract
Résumé

The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices $(a_{α β})$ of order two and a field of symmetric matrices $(b_{α β})$ of order two together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of $R^{2}$ , then there exists an immersion $θ : ω \to R^{3}$ such that these fields are the first and second fundamental forms of the surface $θ (ω)$ and this surface is unique up to proper isometries in $R^{3}$ .

In this Note, we identify new compatibility conditions, expressed again in terms of the functions $a_{α β}$ and $b_{α β}$ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form

\partial_{1} A_{2} - \partial_{2} A_{1} + A_{1} A_{2} - A_{2} A_{1} = 0 in ω,

where

A_{1}

and

A_{2}

are antisymmetric matrix fields of order three that are functions of the fields

(a_{α β})

and

(b_{α β})

, the field

(a_{α β})

appearing in particular through its square root. The unknown immersion

θ : ω \to R^{3}

is found in the present approach in function spaces ‘with little regularity’, viz.,

W_{loc}^{2, p} (ω; R^{3})

,

p > 2

.

Le théorème fondamental de la théorie des surfaces affirme classiquement que, si un champ de matrices $(a_{α β})$ symétriques définies positives d'ordre deux et un champ de matrices $(b_{α β})$ symétriques d'ordre deux satisfont ensemble les équations de Gauss et Codazzi–Mainardi dans un ouvert $ω \subset R^{2}$ connexe et simplement connexe, alors il existe une immersion $θ : ω \to R^{3}$ telle que ces deux champs soient les première et deuxième formes fondamentales de la surface $θ (ω)$ , et cette surface est unique aux isométries propres de $R^{3}$ près.

Dans cette Note, nous identifions de nouvelles conditions de compatibilité, exprimées à nouveau à l'aide des fonctions $a_{α β}$ et $b_{α β}$ , qui conduisent aussi à un théorème analogue d'existence et d'unicité. Ces conditions sont de la forme

\partial_{1} A_{2} - \partial_{2} A_{1} + A_{1} A_{2} - A_{2} A_{1} = 0 dans ω,

où

A_{1}

et

A_{2}

sont des champs de matrices antisymétriques d'ordre trois, qui sont des fonctions des champs

(a_{α β})

et

(b_{α β})

, le champ

(a_{α β})

apparaissant en particulier par l'intermédiaire de sa racine carrée. L'immersion inconnue

θ : ω \to R^{3}

est trouvée dans cette approche dans des espaces fonctionnelles « de faible régularité », à savoir

W_{loc}^{2, p} (ω; R^{3})

,

p > 2

.

Accepted: 2007-07-18
Published online: 2007-08-21

DOI: 10.1016/j.crma.2007.07.014

Author's affiliations:

Philippe G. Ciarlet ¹; Liliana Gratie ²; Cristinel Mardare ³

¹ Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
² Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
³ Laboratoire Jacques-Louis Lions, Université Pierre-et-Marie-Curie, 4, place Jussieu, 75005 Paris, France

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     author = {Philippe G. Ciarlet and Liliana Gratie and Cristinel Mardare},
     title = {New compatibility conditions for the fundamental theorem of surface theory},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {273--278},
     publisher = {Elsevier},
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     number = {5},
     year = {2007},
     doi = {10.1016/j.crma.2007.07.014},
     language = {en},
}

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Philippe G. Ciarlet; Liliana Gratie; Cristinel Mardare. New compatibility conditions for the fundamental theorem of surface theory. Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 273-278. doi : 10.1016/j.crma.2007.07.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.07.014/

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