Comptes Rendus
no. 21-22
Differential Geometry
A Note on surfaces with parallel mean curvature

Presented by: the Editorial Board

Dorel Fetcu1; Harold Rosenberg2
1 Department of Mathematics, “Gh. Asachi” Technical University of Iasi, Bd. Carol I no. 11, 700506 Iasi, Romania
2 IMPA, Estrada Dona Castorina, 110, 22460-320 Rio de Janeiro, Brasil
Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1195-1197.

We use a Simons type equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.

Dans cette Note, on étudie des immersions isométriques de surfaces complètes Σ2 dans Mn(c)×R, ou Mn(c) est une variété complète simplement connexe de courbure sectionnelle constante c. On classifie ces immersions, lorsque leur vecteur courbure moyenne est parallèle dans le fibré normal et leur courbure intrinsèque est positive ou nulle. Lʼoutil principal est une différentielle quadratique holomorphe dont la partie sans trace satisfait lʼéquation de Codazzi.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.10.012

Dorel Fetcu 1; Harold Rosenberg 2

1 Department of Mathematics, “Gh. Asachi” Technical University of Iasi, Bd. Carol I no. 11, 700506 Iasi, Romania
2 IMPA, Estrada Dona Castorina, 110, 22460-320 Rio de Janeiro, Brasil 
@article{CRMATH_2011__349_21-22_1195_0,
     author = {Dorel Fetcu and Harold Rosenberg},
     title = {A {Note} on surfaces with parallel mean curvature},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1195--1197},
     publisher = {Elsevier},
     volume = {349},
     number = {21-22},
     year = {2011},
     doi = {10.1016/j.crma.2011.10.012},
     language = {en},
}
TY  - JOUR
AU  - Dorel Fetcu
AU  - Harold Rosenberg
TI  - A Note on surfaces with parallel mean curvature
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 1195
EP  - 1197
VL  - 349
IS  - 21-22
PB  - Elsevier
DO  - 10.1016/j.crma.2011.10.012
LA  - en
ID  - CRMATH_2011__349_21-22_1195_0
ER  - 
%0 Journal Article
%A Dorel Fetcu
%A Harold Rosenberg
%T A Note on surfaces with parallel mean curvature
%J Comptes Rendus. Mathématique
%D 2011
%P 1195-1197
%V 349
%N 21-22
%I Elsevier
%R 10.1016/j.crma.2011.10.012
%G en
%F CRMATH_2011__349_21-22_1195_0
Dorel Fetcu; Harold Rosenberg. A Note on surfaces with parallel mean curvature. Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1195-1197. doi : 10.1016/j.crma.2011.10.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.012/

[1] U. Abresch; H. Rosenberg A Hopf differential for constant mean curvature surfaces in S2×R and H2×R, Acta Math., Volume 193 (2004), pp. 141-174

[2] U. Abresch; H. Rosenberg Generalized Hopf differentials, Mat. Contemp., Volume 28 (2005), pp. 1-28

[3] H. Alencar; M. do Carmo; R. Tribuzy A theorem of Hopf and the Cauchy–Riemann inequality, Comm. Anal. Geom., Volume 15 (2007), pp. 283-298

[4] H. Alencar; M. do Carmo; R. Tribuzy A Hopf theorem for ambient spaces of dimensions higher than three, J. Differential Geometry, Volume 84 (2010), pp. 1-17

[5] M. Batista, Simons type equation in S2×R and H2×R and applications, Ann. Inst. Fourier, in press.

[6] S.-Y. Cheng; S.-T. Yau Hypersurfaces with constant scalar curvature, Math. Ann., Volume 225 (1977), pp. 195-204

[7] J.M. Espinar; H. Rosenberg Complete constant mean curvature surfaces in homogeneous spaces, Comment. Math. Helv., Volume 86 (2011), pp. 659-674

[8] A. Huber On subharmonic functions and differential geometry in the large, Comment. Math. Helv., Volume 32 (1957), pp. 13-71

[9] K. Nomizu; B. Smyth A formula of Simonsʼ type and hypersurfaces with constant mean curvature, J. Differential Geometry, Volume 3 (1969), pp. 367-377

[10] J. Simons Minimal varieties in Riemannian manifolds, Ann. of Math., Volume 88 (1968), pp. 62-105

[11] B. Smyth Submanifolds of constant mean curvature, Math. Ann., Volume 205 (1973), pp. 265-280

Cited by Sources:

Comments - Policy