Comptes Rendus
Differential Geometry
Liouville type theorem and uniform bound for the Lichnerowicz equation and the Ginzburg–Landau equation
Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 993-996.

In this Note, we prove the Liouville type result for smooth positive solutions to the Lichnerowicz equation in Rn. Using the same method, we also give the uniform bound of the smooth solutions to Ginzburg–Landau equation in the whole space. Similar results on a complete non-compact Riemannian manifold with the Ricci curvature bounded from below are also considered.

Dans cette Note on démontre un résultat de type de Liouville des solutions régulières pour l'équation de Lichnerowicz dans Rn. En utilisant la même méthode on détermine également une borne uniforme inférieure des solutions régulières pour l'équation de Ginzburg–Landau dans tout l'espace. Des résultats analogues sont donnés dans le cas d'une variété riemannienne non compacte complète de courbure de Ricci bornée inférieurement.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2010.07.031

Li Ma 1

1 Department of mathematics, Henan Normal university, Xinxiang, 453007, China
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Li Ma. Liouville type theorem and uniform bound for the Lichnerowicz equation and the Ginzburg–Landau equation. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 993-996. doi : 10.1016/j.crma.2010.07.031. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.031/

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[3] Li Ma; Xingwang Xu Uniform bound and a non-existence result for Lichnerowicz equation in the whole n-space, C. R. Mathematique Ser. I, Volume 347 (2009), pp. 805-808

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[5] W. Shi Deformation the metric on complete Riemannian manifolds, J. Diff. Geom., Volume 30 (1989), pp. 223-301

Cited by Sources:

The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 20090002110019.

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