[Résultats à la Moser dans les espaces produit de Riemann]
Dans cette courte Note, nous obtenons de nouvelles extensions d'un théorème classique de Moser [8] comme application du principe bien connu du maximum généralisé de Omori–Yau. Plus précisément, soit u une fonction lisse définissant un graphe entier, CMC, construit sur une fibre d'un espace produit de Riemann du type . Nous montrons alors que, sous des contraintes convenables sur la norme du gradient de u, cette fonction doit en fait être constante.
In this short paper, as applications of the well-known generalized maximum principle of Omori–Yau, we obtain new extensions of a classical theorem due to Moser [8]. More precisely, under suitable constraints on the norm of the gradient of the smooth function u that defines an entire CMC graph constructed over a fiber of a Riemannian product space of the type , we show that u must actually be constant.
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@article{CRMATH_2015__353_11_1017_0,
author = {Oliveira, Arlandson M.S. and de Lima, Henrique F.},
title = {Moser-type results in {Riemannian} product spaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {1017--1021},
publisher = {Elsevier},
volume = {353},
number = {11},
year = {2015},
doi = {10.1016/j.crma.2015.09.001},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2015.09.001/}
}
TY - JOUR AU - Oliveira, Arlandson M.S. AU - de Lima, Henrique F. TI - Moser-type results in Riemannian product spaces JO - Comptes Rendus. Mathématique PY - 2015 SP - 1017 EP - 1021 VL - 353 IS - 11 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2015.09.001/ DO - 10.1016/j.crma.2015.09.001 LA - en ID - CRMATH_2015__353_11_1017_0 ER -
%0 Journal Article %A Oliveira, Arlandson M.S. %A de Lima, Henrique F. %T Moser-type results in Riemannian product spaces %J Comptes Rendus. Mathématique %D 2015 %P 1017-1021 %V 353 %N 11 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2015.09.001/ %R 10.1016/j.crma.2015.09.001 %G en %F CRMATH_2015__353_11_1017_0
Oliveira, Arlandson M.S.; de Lima, Henrique F. Moser-type results in Riemannian product spaces. Comptes Rendus. Mathématique, Tome 353 (2015) no. 11, pp. 1017-1021. doi : 10.1016/j.crma.2015.09.001. http://www.numdam.org/articles/10.1016/j.crma.2015.09.001/
[1] A Bernstein-type theorem for Riemannian manifolds with a Killing field, Ann. Glob. Anal. Geom., Volume 31 (2007), pp. 363-373
[2] Sur une théorème de géometrie et ses applications aux équations dérivées partielles du type elliptique, Commun. Soc. Math. Kharkov, Volume 15 (1914), pp. 38-45
[3] Minimal cones and the Bernstein problem, Invent. Math., Volume 7 (1969), pp. 243-268
[4] Complete vertical graphs with constant mean curvature in semi-Riemannian warped products, Bull. Belg. Math. Soc. Simon Stevin, Volume 16 (2009), pp. 91-105
[5] Una estensione del teorema di Bernstein, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 19 (1965), pp. 79-85
[6] On the oriented Plateau problem, Rend. Circ. Mat. Palermo, Volume 11 (1962), pp. 69-90
[7] On S. Bernstein's theorem on surfaces of nonpositive curvature, Proc. Amer. Math. Soc., Volume 1 (1950), pp. 80-85
[8] On Harnack's theorem for elliptic differential equations, Commun. Pure Appl. Math., Volume 14 (1961), pp. 577-591
[9] Isometric immersions of Riemannian manifolds, J. Math. Soc. Jpn., Volume 19 (1967), pp. 205-214
[10] Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983
[11] Minimal surfaces in , Ill. J. Math., Volume 46 (2002), pp. 1177-1195
[12] The half-space property and entire positive minimal graphs in , J. Differ. Geom., Volume 95 (2013), pp. 321-336
[13] Graphs with parallel mean curvature, Proc. Amer. Math. Soc., Volume 107 (1989), pp. 449-458
[14] Minimal varieties in Riemannian manifolds, Ann. Math., Volume 88 (1968), pp. 62-105
[15] Harmonic functions on complete Riemannian manifolds, Commun. Pure Appl. Math., Volume 28 (1975), pp. 201-228
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