It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian manifolds.

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Abstract. It is still an open question whether a compact embedded hypersurface in the Euclidean space R n+1 with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of surfaces in R 3 . In a recent paper [3] the first and third authors have shown that this is true for the case of hypersurfaces in R n+1 with constant scalar curvature, and more generally, hypersurfaces with constant higher order r-mean curvature, when r ≥ 2. In this paper we deal with some aspects of the classical problem above, by considering it in a more general context. Specifically, our starting general ambient space is an orientable Riemannian manifold M , where we will consider a general geometric configuration consisting of an immersed hypersurface into M with boundary on an oriented hypersurface P of M . For such a geometric configuration, we study the relationship between the geometry of the hypersurface along its boundary and the geometry of its boundary as a hypersurface of P , as well as the geometry of P as a hypersurface of M. Our approach allows us to derive, among others, interesting results for the case where the ambient space has constant curvature (the Euclidean space R n+1 , the hyperbolic space H n+1 , and the sphere S n+1 ). In particular, we are able to extend the symmetry results given in [3] to the case of hypersurfaces with constant higher order r-mean curvature in the hyperbolic space and in the sphere.

Abstract. The subject of this paper is properly embedded H−surfaces in Riemannian three manifolds of the form M 2 × R, where M 2 is a complete Riemannian surface. When M 2 = R 2 , we are in the classical domain of H−surfaces in R 3 . In general, we will make some assumptions about M 2 in order to prove stronger results, or to show the effects of curvature bounds in M 2 on the behavior of H−surfaces in M 2 × R.