HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN A HYPERBOLIC SPACE

doi:10.1016/S0252-9602(11)60300-7

Abstract

Abstract:

Let Mⁿbe 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 S^k(r)×H^^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 Mⁿis isometric to the Riemannian product Sⁿ^-1(r)×H¹(-1/(r²+ρ²)) or S¹(r)×Hⁿ^-1(-1/(r²+ρ²)), 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 Mⁿor (iii) (n-1)t²₂+c²t^-2₂≤S≤ (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

CLC Number:

53C42

SU Bian-Ping, SHU Shi-Chang, Yi Annie Han. HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN A HYPERBOLIC SPACE[J].Acta mathematica scientia,Series B, 2011, 31(3): 1091-1102.

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