Isometric immersions between spaces forms

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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.