伪黎曼乘积空间中具有平行平均曲率向量的曲面

数学物理学报 ›› 2016, Vol. 36 ›› Issue (6): 1027-1039.

伪黎曼乘积空间中具有平行平均曲率向量的曲面

邱望华¹, 侯中华²

1. 九江学院理学院江西九江 332005;
2. 大连理工大学数学科学学院辽宁大连 116024

收稿日期:2016-02-24 修回日期:2016-07-31 出版日期:2016-12-26 发布日期:2016-12-26
通讯作者: 侯中华 E-mail:zhhou@dlut.edu.cn
基金资助:
国家自然科学基金（61473059）资助

Surfaces in Products of Semi-Riemannian Space Forms with Parallel Mean Curvature Vector

Qiu Wanghua¹, Hou Zhonghua²

1. College of Sciences, Jiujiang University, Jiangxi Jiujiang 332005;
2. Institute of Mathematics, Dalian University of Technology, Liaoning Dalian 116024

Received:2016-02-24 Revised:2016-07-31 Online:2016-12-26 Published:2016-12-26
Supported by:
Supported by the NSFC (61473059)

摘要/Abstract

摘要：

Batista在M²（c）×R中的具有常平均曲率的曲面上引入了一个特殊的（1，1）型张量S.之后，Fetcu和Rosebberg将张量S推广到Mⁿ（c）×R中的具有平行平均曲率向量的曲面上.该文将张量S推广到了伪黎曼乘积空间中的曲面上，并研究了S的Pinching问题，得到了若干Pinching常数.特别地，对外围空间是黎曼乘积空间的情况，得到的Pinching常数优于Batista得到的相应的Pinching常数.

关键词: 平行平均曲率向量, 乘积空间, Simons型方程, Pinching常数

Abstract:

Batista introduced a special (1,1) tensor S on a CMC immersed surface Σ² in M²(c)×R. Later on, Fetcu and Rosebberg extended (1,1) tensor S to PMC surface Σ²M^n-1(c)×R. In the present paper, the authors consider a more general tensor S on PMC immersed surface Σ² in Lorentzian product spaces (M^n-1(c)×R, g_-1) and Riemmannian product spaces (M^n-1(c)×R, g₊₁). The authors compute the Simons type equations of |S|², and characterize CMC surfaces in (M²(c)×R, g_ε). For case ε=+1, we obtain several pinching constants greater than that given by Batista.

Key words: Parallel mean curvature vector, Simons type equation, Product spaces, Pinching constants

中图分类号:

O186.12

邱望华, 侯中华. 伪黎曼乘积空间中具有平行平均曲率向量的曲面[J]. 数学物理学报, 2016, 36(6): 1027-1039.

Qiu Wanghua, Hou Zhonghua. Surfaces in Products of Semi-Riemannian Space Forms with Parallel Mean Curvature Vector[J]. Acta mathematica scientia,Series A, 2016, 36(6): 1027-1039.

参考文献

[1] Abresch U, Rosenberg H. A Hopf differential for conastant mean curvature surfaces in S²×R and H²×R. Acta Math, 2004, 193:141-174
[2] Alencar H, do Carmo M, Tribuzy R. A theorem of Hopf and the Cauchy-Riemann inequality. Comm Anal Geom, 2007, 15:283-298
[3] Alencar H, do Carmo M, Isabel F, Tribuzy R. A theorem of Hopf and the Cauchy-Riemann inequality II. Bull Braz Math Soc, 2007, 38:525-532
[4] Alencar H, do Carmo M, Tribuzy R. A Hopf theorem for ambient spaces of dimensions higher than three. J Diff Geom, 2010, 84:1-17
[5] Alencar H, do Carmo M, Tribuzy R. Surfaces of M_κ²×R invariant under a one-parameter group of isometries. Ann Mat Pura Appl, 2004, 193(4):517-527
[6] Aquino C P, de Lima H F, Lima Eraldo A. On the angle of complete CMC hypersurfaces in Riemannian product spaces. Diff Geom Appl, 2014, 33:139-148
[7] Baek J O, Cheng Q M, Suh Y J. Complete space-like hypersurfaces in locally symmetric Lorentz spaces. J Geom and Phys, 2004, 49:231-247
[8] Batista M. Simons type equation in S²×R and H²×R and applications. Ann Inst Fourier (Grenoble), 2011, 61:1299-1322
[9] do Carmo M. Some recent developments on Hopf's holomorphic form. Results Math, 2011, 60:175-183
[10] do Carmo M, Isabel F. A Hopf theorem for open surfaces in product spaces. Forum Math, 2009, 21:951-963
[11] Chen H, Chen G Y, Li H Z. Some pinching theorems for minimal submanifolds in S^m(1)×R. Sci China Math, 2013, 56:1679-1688
[12] Cheng S Y, Yau S T. Hypersurfaces with constant scalar curvature. Math Ann, 1997, 225:195-204
[13] Deng Q T. Complete hypersurfaces with constant mean curvature and finite index in hyperbolic spaces. Acta Math Sci, 2011, 31:353-360
[14] Dillen F, Fastenakels J, Van der Veken J. Surfaces in S²×R with a canonical principal direction. Ann Global Anal Geom, 2009, 35:381-396
[15] Dillen F, Fastenakels J, Van der Veken J, Vrancken L. Constant angle surfaces in S²×R. Monaths Math Soc, 2007, 152(2):89-96
[16] Dillen F, Munteanu M I. Constant angle surfaces in H²×R. Bull Braz Math Soc, 2009, 40(1):85-97
[17] Dillen F, Munteanu M I, Nistor A I. Canonical coordinates and principal directions for surfaces in H²×R. Taiwanese J Math, 2011, 15:2265-2289
[18] Espinar J M, Rosenberg H. Complete constant mean curvature surfaces in homogeneous spaces. Comment Math Helv, 2011, 86:659-674
[19] Ferreira M J, Tribuzy R. Parallel mean curvature surfaces in symmetric spaces. Ark Mat, 2014, 52:93-98
[20] Fetcu D, Rosenberg H. Surfaces with parallel mean curvature in S³×R and H³×R. Mich Math J, 2012, 61:715-729
[21] Fetcu D, Rosenberg H. On complete submanifolds with parallel mean curvature in product spaces. Rev Mat Iberoam, 2013, 29:1283-1306
[22] Fetcu D, Rosenberg H. A note on Surfaces with parallel mean curvature. C R Acad Sci Paris Ser I, 2011, 349:1195-1197
[23] Fu Y, Nistor A I. Constant angle property and canonical principal directions for surfaces in M²(c)×R₁. Mediterr J Math, 2013, 10:1035-1049
[24] Hou Z H, Qiu W H. Submanifolds with parallel mean curvature vector field in product spaces. Vietnam J Math, 2015, 43:705-724
[25] Hou Z H, Qiu W H. A classification theorem for complete PMC surfaces with non-negative Gaussian curvature in Mⁿ×R. Taiwanese J Math, 2016, 20:205-226
[26] de Lima H F, Lima Eraldo A. Generalized maximum principles and the unicity of complete spacelike hypersurfaces immersed in a Lorentzian product space. Beitr Algebra Geom, 2014, 55:59-75
[27] Su B P, Shu S C, Han Y A. Hypersurfaces with constant mean curvature in a hyperbolic space. Acta Math Sci, 2011, 31:1091-1102