Isometric immersions between spaces forms

黎曼流形M^^n ^^<n+1>中具有常平均曲率的超曲面M^n是变分问题的解，该变分问题是在某些变分下使面积函数最小化;容许变分仅是那些使某个体积函数保持不变的变分。在欧氏（n+1）空间R^<n+1>和欧氏（n+1）球面S^<n+1>中存在许多常平均曲率的完备超曲面，但在双曲（n+1）-空间H^<n+1>中，除了脐超曲面外，关于这类超曲面的结果很少。本文的第一个主要目的是构造H^<n+1>中三种不同类型的单参数常平均曲率旋转超曲面族.巴博萨，do Carmo和Schenburg定义了黎曼流形M^n中常平均曲率超曲面M^n的稳定性概念.当M^2是完备的和非紧的时，da Silveira处理了这种情况。巴博萨、多·卡尔莫（do Carmo）和埃森伯格（Eschenburg）讨论了M^n是紧的情形。Luo讨论了R^<n+1>中具有常平均曲率的完备非紧超曲面的稳定性，除了H=0的情形外，关于H^<n+1>中具有常平均曲率H的完备非紧黎曼超曲面的稳定性知之甚少，当3 ≤ n.本文的第二个主要目的是讨论H^<n+1>中具有常平均曲率H的超曲面的稳定性。

Hypersurfaces M^n with constant mean curvature in a Riemannian manifold M^^〜^<n+1> are solutions to the variational problem of minimizing the area function for certain variations ; the admissible variations are only those that leave a certain volume function fixed. This isoperimetric character of the variational problem associated to hypersurfaces with constant mean curvature introduces additional complications in the treatment of stability of such hypersurfaces.There are many complete hypersurfaces with constant mean curvature in Euclidean (n+1)-space R^<n+1> and Euclidean (n+1)-sphere S^<n+1>, but in the hyperbolic (n+1)-space H^<n+1> there have been few results on such hypersurfaces except umbilical ones. First main purpose of this paper is to construct one-parameter families of three distinct type, rotation hypersurfaces with constant mean curvature in H^<n+1>, explicitly.Barbosa, do Carmo and Eschenburg have defined the notion of stability for hypersurfaces M^n with constant mean curvature in a Riemannian manifold M^^〜^<n+1>. The case where M^2 is complete and noncompact is treated by da Silveira. The case where M^n, is compact is treated by Barbosa, do Carmo and Eschenburg. Luo has discussed the stability of complete noncompact hypersurfaces with constant mean curvature in R^<n+1>.Except for the case where H=0 very little is known about stability of complete and noncompact Riemannian hypersurfaces of H^<n+1> with constant mean curvature H, when 3【less than or equal】n. Second main purpose of this paper is to discuss the stability of the hypersurfaces in H^<n+1> with constant mean curvature H.