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

We consider compact surfaces with constant nonzero mean curvature whose boundary is a convex planar Jordan curve. We prove that if such a surface is orthogonal to the plane of the boundary, then it is a hemisphere. and we show that, in the above conditions, if M is embedded and is convex, then M is a hemisphere. Explicitly we prove that:

In this work we deal with surfaces immersed in R 3 with constant mean curvature and circular boundary. We improve some global estimates for area and volume of such immersions obtained by other authors. We still establish the uniqueness of the spherical cap in some classes of cmc surfaces.

In this work we will deal with disc type surfaces of constant mean curvature in the three dimensional hyperbolic space which are given as graphs of smooth functions over planar domains. From the various types of graphs that could be defined in the hyperbolic space we consider in particular the horizontal and the geodesic graphs. We proved that if the mean curvature is constant, then such graphs are equivalent in the following sense: suppose that M is a constant mean curvature surface in the 3-hyperbolic space such that M is a geodesic graph of a function ρ that is zero at the boundary, then there exist a smooth function f, that also vanishes at the boundary, such that M is a horizontal graph of f. Moreover, the reciprocal is also true.