We study stable capillary surfaces in a euclidean ball in the absence of gravity. We prove, in particular, that such a surface must be a flat disk or a spherical cap if it has genus zero. We also prove that its genus is at most one and it has at most three connected boundary components in case it is minimal. Some of our results also hold in H 3 and S 3 .

We obtain an estimate for the norm of the second fundamental form of stable H-surfaces in Riemannian 3-manifolds with bounded sectional curvature. Our estimate depends on the distance to the boundary of the surface and on the bound on the sectional curvature but not on the manifold itself. We give some applications, in particular we obtain an interior gradient estimate for H-sections in Killing submersions.

Abstract. We discuss existence and classification of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We classify such surfaces in H 2 R, S 2 R and the Sol group. We prove nonexistence in the Berger spheres and in the remaining model geometries other than the space forms. Mathematics Subject Classification (2000). 53C30, 53B25.

We prove that a complete embedded maximal surface in L 3 with a finite number of singularities is an entire maximal graph with conelike singularities over any spacelike plane, and so, it is asymptotic to a spacelike plane or a half catenoid.We show that the moduli space Gn of entire maximal graphs over {x3 = 0} in L 3 with n + 1 ≥ 2 singular points and vertical limit normal vector at infinity is a 3n + 4-dimensional differentiable manifold. The convergence in Gn means the one of conformal structures and Weierstrass data, and it is equivalent to the uniform convergence of graphs on compact subsets of {x3 = 0}. Moreover, the position of the singular points in R 3 and the logarithmic growth at infinity can be used as global analytical coordinates with the same underlying topology. We also introduce the moduli space Mn of marked graphs with n + 1 singular points (a mark in a graph is an ordering of its singularities), which is a (n + 1)-sheeted covering of Gn. We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient spacê Mn is an analytic manifold of dimension 3n−1. This manifold can be identified with a spinorial bundle Sn associated to the moduli space of Weierstrass data of graphs in Gn. Notations and Preliminary results .Since M is spacelike, then |g| = 1 on M.Remark 2.1 For convenience, we also deal with surfaces M having ∂(M ) = ∅, and in this case, we always suppose that φ 3 and g extend analitically beyond ∂M.Conversely, let M, g and φ 3 be a Riemann surface with possibly non empty boundary, a meromorphic map on M and an holomorphic 1-form φ 3 on M, such that |g(P )| = 1, ∀P ∈ M, and the 1-forms φ j , j = 1, 2, 3 defined as above are holomorphic, have no real periods and have no common zeroes.

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