数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (3): 1091-1102.doi: 10.1016/S0252-9602(11)60300-7

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HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN A HYPERBOLIC SPACE

苏变萍, 舒世昌*, Yi Annie Han   

  1. Department of Science, Xi'an University of Architecture and Technology, Xi'an 710055, China|Department of Mathematics, Xianyang Normal University, Xianyang 712000, China
  • 收稿日期:2008-12-16 出版日期:2011-05-20 发布日期:2011-05-20
  • 基金资助:

    Project supported by NSF of Shaanxi Province (SJ08A31), NSF of Shaanxi Educational Committee (2008JK484; 2010JK642) and Talent Fund of Xi'an University of Architecture and Technology.

HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN A HYPERBOLIC SPACE

 SU Bian-Ping, SHU Shi-Chang*, Yi Annie Han   

  1. Department of Science, Xi'an University of Architecture and Technology, Xi'an 710055, China|Department of Mathematics, Xianyang Normal University, Xianyang 712000, China
  • Received:2008-12-16 Online:2011-05-20 Published:2011-05-20
  • Supported by:

    Project supported by NSF of Shaanxi Province (SJ08A31), NSF of Shaanxi Educational Committee (2008JK484; 2010JK642) and Talent Fund of Xi'an University of Architecture and Technology.

摘要:

Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space Hn+1(c) with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that (1) if the multiplicities of the two distinct
principal curvatures are greater than 1, then Mn is isometric to the Riemannian product Sk(rH^n-k(-1/(r^2+ρ2)), where r>0 and 1<k<n-1; (2) if H2>-c and one of the two distinct principal curvatures is simple, then Mn is isometric to the Riemannian product Sn-1(rH1(-1/(r22)) or S1(rHn-1(-1/(r22)), r>0, if one of the following conditions is satisfied (i) S≤(n-1)t22+c2t-22 on Mn or (ii) S≥(n-1)t21+c2t-21 on Mn or (iii) (n-1)t22+c2t-22S≤ (n-1)t21+c2t-21 on Mn, where t1 and t2 are the positive real roots of (1.5).

关键词: hypersurface, hyperbolic space, scalar curvature, mean curvature, principal curvature

Abstract:

Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space Hn+1(c) with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that (1) if the multiplicities of the two distinct
principal curvatures are greater than 1, then Mn is isometric to the Riemannian product Sk(rH^n-k(-1/(r^2+ρ2)), where r>0 and 1<k<n-1; (2) if H2>-c and one of the two distinct principal curvatures is simple, then Mn is isometric to the Riemannian product Sn-1(rH1(-1/(r22)) or S1(rHn-1(-1/(r22)), r>0, if one of the following conditions is satisfied (i) S≤(n-1)t22+c2t-22 on Mn or (ii) S≥(n-1)t21+c2t-21 on Mn or (iii) (n-1)t22+c2t-22S≤ (n-1)t21+c2t-21 on Mn, where t1 and t2 are the positive real roots of (1.5).

Key words: hypersurface, hyperbolic space, scalar curvature, mean curvature, principal curvature

中图分类号: 

  • 53C42