A Note on the Bernstein property of a fourth order complex partial differential equations

Saïd Asserda

doi:10.1016/j.crma.2011.11.016

Partial Differential Equations/Differential Geometry

A Note on the Bernstein property of a fourth order complex partial differential equations

Presented by: the Editorial Board

Saïd Asserda ¹

¹ Laboratoire dʼanalyse mathématique et systèmes informatiques, université Ibn Tofail, faculté des sciences, BP 133, Kénitra, Morocco

Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 41-44.

Abstract
Résumé

For a smooth strictly plurisubharmonic function u on an open set $Ω \subset C^{n}$ and F a $C^{1}$ nondecreasing function on $R_{+}^{⁎}$ , we investigate the complex partial differential equations

Δ_{g} \log \det (u_{i \bar{j}}) = F (\det (u_{i \bar{j}})) ‖ \nabla_{g} \log \det (u_{i \bar{j}}) ‖_{g}^{2},

where

Δ_{g}

,

‖ . ‖_{g}

and

\nabla_{g}

are the Laplacian, tensor norm and the Levi-Civita connexion, respectively, with respect to the Kähler metric

g = \partial \bar{\partial} u

. We show that the above PDEʼs has a Bernstein property, i.e.

\det (u_{i \bar{j}})

is constant on Ω, provided that g is complete, the Ricci curvature of g is bounded below and F satisfies

\inf_{t \in R^{+}} (2 t F^{'} (t) + \frac{F {(t)}^{2}}{n}) > \frac{1}{4}

and

F (\max_{B (R)} \det u_{i \bar{j}}) = o (R)

.

Pour une fonction u strictement plurisouharmonique de classe $C^{\infty}$ sur un ouvert Ω de $C^{n}$ et F une fonction de classe $C^{1}$ croissante sur $R^{+}$ , on considère lʼéquation aux dérivées partielles complexes

Δ_{g} \log \det (u_{i \bar{j}}) = F (\det (u_{i \bar{j}})) ‖ \nabla_{g} \log \det (u_{i \bar{j}}) ‖_{g}^{2},

où

Δ_{g}

,

‖ . ‖_{g}

et

\nabla_{g}

sont respectivement le Laplacian, la norme et la connexion de Levi-Civita par rapport à la métrique Kählerienne

g = \partial \bar{\partial} u

. On montre que lʼEDP précédente vérifie la propriété de Bernstein, i.e.

\det (u_{i \bar{j}})

est constante sur Ω, pourvu que g soit complète, la courbure de Ricci de g soit minorée et F satisfasse

\inf_{t \in R^{+}} (2 t F^{'} (t) + \frac{F {(t)}^{2}}{n}) > \frac{1}{4}

et

F (\max_{B (R)} \det u_{i \bar{j}}) = o (R)

.

Received: 2011-04-21
Accepted: 2011-11-24
Published online: 2011-12-08

DOI: 10.1016/j.crma.2011.11.016

Author's affiliations:

Saïd Asserda ¹

¹ Laboratoire dʼanalyse mathématique et systèmes informatiques, université Ibn Tofail, faculté des sciences, BP 133, Kénitra, Morocco

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     author = {Sa{\"\i}d Asserda},
     title = {A {Note} on the {Bernstein} property of a fourth order complex partial differential equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {41--44},
     publisher = {Elsevier},
     volume = {350},
     number = {1-2},
     year = {2012},
     doi = {10.1016/j.crma.2011.11.016},
     language = {en},
}

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Saïd Asserda. A Note on the Bernstein property of a fourth order complex partial differential equations. Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 41-44. doi : 10.1016/j.crma.2011.11.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.11.016/

References
Cited by

[1] G. Chen; L. Sheng A Bernstein property of a class of fourth order complex partial differential equations, Results Math., Volume 58 (2010), pp. 81-92

[2] R.E. Greene; H. Wu Function Theory on Manifolds Which Possess a Pole, Lecture Notes in Math., vol. 699, 1979

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