Uniqueness of horospheres and geodesic cylinders in hyperbolic space

A hyperbolic analogon to Hartman’s characterization of orthogonal sphere cylinders is proved: Let Mn ? Hⁿ⁺¹ be a noncompact closed hypersurface with sectional curvature K ≥ 0 which bounds a convex set. Assume further H_r ≡ c for one normalized mean curvature. Then Mⁿ is a horosphere or a geodesic cylinder if $r{\leq}\ {2\over 3}\ (n+1)$ . For $r >\ {2\over 3}\ (n+1)$ the same follows but only if c lies in a specified interval which however covers the case of a horosphere. The argumentation is based on results of S.B. Alexander and R.B. Currier on the infinity set of certain convex hypersurfaces, the comparison with interior spindle surfaces, first eigenvalue estimates for Voss operators and variational properties of relevant curvature expressions.     

Characterizing horospheres of the hyperbolic space via higher order mean curvatures

In this paper we establish new characterization results concerning horospheres of the hyperbolic space under certain appropriate constraints in the behavior of the higher order mean curvatures. Our approach is based on a suitable maximum principle for complete Riemannian manifolds. Moreover, we present examples showing the importance of the main hypothesis of our results.     

Hypersurfaces in simply connected space forms

Let M be a hypersurface in a simply connected space form . We prove some rigidity results for M in terms of lower bounds on the Ricci curvature of the hypersurface M.     

Hypersurfaces of restricted type in Minkowski space

A submanifold M _n ^r of Minkowski space is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of to the tangent space of M _n ^r at every point of M _n ^r . In this paper we completely classify hypersurfaces of restricted type in . More precisely, we prove that a hypersurface of is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S ^k × , S _k ¹ × , H ^k × , S _n ¹ , H ⁿ , with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium.     

Hypersurfaces of a Lorentz space form

The project supported by NNSFC, FECC and CPF.     

Hypersurfaces in hyperbolic space with support function

Based on [19], we develop a global correspondence between immersed hypersurfaces ?:Mⁿ→Hⁿ⁺¹$?:{\mathrm{M}}^n\to {\mathbb{H}}^{n+1}$ satisfying an exterior horosphere condition, also called here horospherically concave hypersurfaces, and complete conformal metrics e^2ρ_gSn$e^{2\rho }{g}_{{\mathbb{S}}^n}$ on domains Ω in the boundary Sⁿ${\mathbb{S}}^n$ at infinity of Hⁿ⁺¹${\mathbb{H}}^{n+1}$, where ρ   is the horospherical support function, _∂∞?(Mⁿ)=∂Ω${\partial }_{\infty }?({\mathrm{M}}^n)=\partial \mathrm{\Omega }$, and Ω is the image of the Gauss map G:Mⁿ→Sⁿ$G:{\mathrm{M}}^n\to {\mathbb{S}}^n$. To do so we first establish results on when the Gauss map G:Mⁿ→Sⁿ$G:{\mathrm{M}}^n\to {\mathbb{S}}^n$ is injective. We also discuss when an immersed horospherically concave hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles for elliptic problems of both immersed hypersurfaces in Hⁿ⁺¹${\mathbb{H}}^{n+1}$ and conformal metrics on domains in Sⁿ${\mathbb{S}}^n$. Consequently, we are able to obtain, for instance, a strong Bernstein theorem for a complete, immersed, horospherically concave hypersurface in Hⁿ⁺¹${\mathbb{H}}^{n+1}$ of constant mean curvature.     

Hypersurfaces of the hyperbolic space with constant scalar curvature

We classify hypersurfaces of the hyperbolic space ?ⁿ⁺¹(c) with constant scalar curvature and with two distinct principal curvatures. Moreover, we prove that if Mⁿ is a complete hypersurfaces with constant scalar curvature n(n ? 1) R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n? 1, then R ≥ c. Additionally, we prove two rigidity theorems for such hypersurfaces.     

Prescribed Mean Curvature Hypersurfaces in Hn+1(-1) with Convex Planar Boundary, I

We study immersed prescribed mean curvature compact hypersurfaces with boundary in Hⁿ⁺¹(-1). When the boundary is a convex planar smooth manifold with all principal curvatures greater than 1, we solve a nonparametric Dirichlet problem and use this, together with a general flux formula, to prove a parametric uniqueness result, in the class of all immersed compact hypersurfaces with the same boundary. We specialize this result to a constant mean curvature, obtaining a characterization of totally umbilic hypersurface caps.     

Hypersurfaces with constant mean curvature in a hyperbolic space

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 S^k（r）×H^n-k（-1/（r² + 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 S^n-1（r） × H¹（-1/（r² +ρ²）） or S¹（r） × H^n-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 Mⁿ or（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）.     

Hypersurfaces with constant mean curvature in hyperbolic space form

In this article, we prove the following theorem: A complete hypersurface of the hyperbolic space form, which has constant mean curvature and non-negative Ricci curvature Q, has non-negative sectional curvature. Moreover, if it is compact, it is a geodesic distance sphere; if its soul is not reduced to a point, it is a geodesic hypercylinder; if its soul is reduced to a point p, its curvature satisfies Q<, and the geodesic spheres centered at p are convex, then it is a horosphere.A part of this work has been done when the second author visited Université Claude Bernard Lyon 1, and was supported by a grant of the People's Republic of China.     

Hypersurfaces whose tangent geodesics do not cover the ambient space

Let be an immersion of an -dimensional connected manifold in an -dimensional connected complete Riemannian manifold without conjugate points. Assume that the union of geodesics tangent to does not cover . Under these hypotheses we have two results. The first one states that is simply connected provided that the universal covering of is compact. The second result says that if is a proper embedding and is simply connected, then is a normal graph over an open subset of a geodesic sphere. Furthermore, there exists an open star-shaped set such that is a manifold with the boundary .