Comptes Rendus
no. 1
Differential geometry
A Bernstein theorem for affine maximal-type hypersurfaces
[Un théorème de Bernstein pour les hypersurfaces de type affine maximal]

Présenté par : the Editorial Board

Shi-Zhong Du1 ; Xu-Qian Fan2
1 Department of Mathematics, Shantou University, Shantou, 515063, PR China
2 Department of Mathematics, Jinan University, Guangzhou, 510632, PR China
Comptes Rendus. Mathématique, Volume 357 (2019) no. 1, pp. 66-73.

Nous obtenons, en toute dimension N et pour un large spectre de valeurs θ, un théorème de Bernstein pour l'équation différentielle partielle d'ordre quatre, de type affine maximal

uijDijw=0,w=[det⁡D2u]−θ
sous l'hypothèse de complétude de la métrique de Calabi. Ceci contient les résultats de Li–Jia [A.M. Li, F. Jia, Ann. Glob. Anal. Geom. 23 (2003)] pour les équations affines maximales et de Zhou [B. Zhou, Calc. Var. Partial Differ. Equ. 43 (2012)] pour l'équation d'Abreu. En particulier, nous généralisons les résultats de Zhou pour 2≤N≤4 à 2≤N≤5.

We obtain, in any dimension N and for a large range of values of θ, a Bernstein theorem for the fourth-order partial differential equation of affine maximal type

uijDijw=0,w=[det⁡D2u]−θ
assuming the completeness of Calabi's metric. This contains the results of Li–Jia [A.M. Li, F. Jia, Ann. Glob. Anal. Geom. 23 (2003)] for affine maximal equations and of Zhou [B. Zhou, Calc. Var. Partial Differ. Equ. 43 (2012)] for Abreu's equation. In particular, we extend the result of Zhou from 2≤N≤4 to 2≤N≤5.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.11.011
Shi-Zhong Du 1 ; Xu-Qian Fan 2

1 Department of Mathematics, Shantou University, Shantou, 515063, PR China
2 Department of Mathematics, Jinan University, Guangzhou, 510632, PR China 
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Shi-Zhong Du; Xu-Qian Fan. A Bernstein theorem for affine maximal-type hypersurfaces. Comptes Rendus. Mathématique, Volume 357 (2019) no. 1, pp. 66-73. doi : 10.1016/j.crma.2018.11.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.11.011/

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