Volume 43 – 2012 – XVII School of Differencial Geometry

We deal with complete linear Weingarten hypersurfaces immersed in a locally symmetric Lorentz space, whose sectional curvatures are supposed to obey some standard controls. In this setting, under suitable boundedness on the norm of the traceless part of the second fundamental form, we are able to show that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures one of which is simple

We prove an equivariant deformation result for Hamiltonian stationary Lagrangian submanifolds of a K¨ahler manifold, with respect to deformations of its metric and almost complex structure that are compatible with an isometric Hamiltonian group action. This yields existence of Hamiltonian stationary Lagrangian submanifolds in possibly non-Kähler symplectic manifolds whose metric is arbitrarily close to a Kähler metric

In this paper we study the existence and non-existence of vertical and horizontal graphs in the space PSL2(R, τ ). We prove that there is no entire horizontal minimal graph. On the other hand, there are entire vertical minimal graphs having prescribed continuous boundary values.

In this paper we construct a harmonic diffeomorphism from C* onto the quotient of the hyperbolic plane H2 by the group < ψ > generated by some parabolic or hyperbolic isometry ψ. It draws its inspiration to a large extent from [CR]. The proof we give here is based on the theory of minimal surfaces: we construct an entire minimal graph Σ c (H/ < ψ >) × R over H/ < ψ >, such that Σ is conformally C*. This solves the problem since the vertical projection from Σ onto H/ < ψ > is a harmonic diffeomorphism when Σ is minimal

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in Hn+1 satisfying f(κ) = σ ∈ (0, 1) with a prescribed asymptotic boundary Γ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if Γ is (Euclidean) starshaped, the solution is unique and also (Euclidean) starshaped while if Γ is mean convex the solution is unique. We also show via a strong duality theorem that analogous results hold in De Sitter space. A novel feature of our approach is a “global interior curvature estimate”