STARSHAPED COMPACT HYPERSURFACES WITH PRESCRIBED m−TH MEAN CURVATURE IN HYPERBOLIC SPACE
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چکیده
Let S be the unit sphere in the Euclidean space R, and let e be the standard metric on S induced from R. Suppose that (u, ρ) are the spherical coordinates in R, where u ∈ S, ρ ∈ [0,∞). By choosing the smooth function φ(ρ) := sinh ρ on [0,∞) we can define a Riemannian metric h on the set {(u, ρ) : u ∈ S, 0 ≤ ρ < ∞} as follows h = dρ + φ(ρ)e. This gives the space form R(−1) which is the hyperbolic space H with sectional curvature −1. For a smooth hypersurface M in R(−1), we denote by λ1, · · · , λn its principal curvatures with respect to the metric g := h|M. Then, for each 1 ≤ k ≤ n, the k-th mean curvature of M is defined as
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