Barbara Nelli

Some Remarks on Embedded Hypersurfaces in Hyperbolic Space of Constant Curvature and Spherical Boundary

We consider embedded hypersurfacesM in hyperbolic space with compact boundaryC and somerth mean curvature functionHr a positive constant. We investigate when symmetries ofC are symmetries ofM. We prove that if 0≤Hr≤1 andC is a sphere thenM is a part of an equidistant sphere. Forr=1 (H1 is the mean curvature) we obtain results whenC is convex.

Minimal graphs in Nil_3: existence and non-existence results

We study the minimal surface equation in the Heisenberg space,

Erratum to “Minimal Surfaces in ℍ2 × ℝ”

Nil_3.

“Minimal Surfaces in ℍ2 × ℝ”

Nil3. A geometric proof of non existence of minimal graphs over non convex, bounded and unbounded domains is achieved for some prescribed boundary data (our proof holds in the Euclidean space as well). We solve the Dirichlet problem for the minimal surface equation over bounded and unbounded convex domains, taking bounded, piecewise continuous boundary value. We are able to construct a Scherk type minimal surface and we use it as a barrier to construct non trivial minimal graphs over a wedge of angle

UNIQUENESS OF H-SURFACES IN H 2 × R, |H| ≤ 1/2, WITH BOUNDARY ONE OR TWO PARALLEL HORIZONTAL CIRCLES

\theta \in [\frac{\pi }{2}, \pi [,

Stable constant mean curvature hypersurfaces

θ∈[π2,π[, taking non negative continuous boundary data, having at least quadratic growth. In the case of an half-plane, we are also able to give solutions (with either linear or quadratic growth), provided some geometric hypothesis on the boundary data are satisfied. Finally, some open problems arising from our work, are posed.