Constant mean curvature surfaces in Sol with non-empty boundary

In homogenous space Sol we study compact surfaces with constant mean curvature and with non-empty boundary. We ask how the geometry of the boundary curve imposes restrictions over all possible configurations that the surface can adopt. We obtain a flux formula and we establish results that assert that, under some restrictions, the symmetry of the boundary is inherited into the surface.Comment: 14 pages; corrected typo

The Dirichlet problem for the α\alpha-singular minimal surface equation

Let \Omega\subset\r^n be a bounded mean convex domain. If $\alpha<0$, we prove the existence and uniqueness of classical solutions of the Dirichlet problem in $\Omega$ for the $\alpha$-singular minimal surface equation with arbitrary continuous boundary data.Comment: 12 pages. Accepted to be published in Archiv der Mathemati

Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space

We review part of the classical theory of curves and surfaces in $3$-dimensional Lorentz-Minkowski space. We focus in spacelike surfaces with constant mean curvature pointing the differences and similarities with the Euclidean space.Comment: Notes of a Mini-Course taught at Instituto de Matematica e Estatistica. University of Sao Paulo, Brazil. The version v2 is a complete revision of version v

Minimal surfaces in Euclidean space with a log-linear density

We study surfaces in Euclidean space ${\mathbb R}^3$ that are minimal for a log-linear density $\phi(x,y,z)=\alpha x+\beta y+\gamma y$, where $\alpha,\beta,\gamma$ are real numbers not all zero. We prove that if a surface is $\phi$-minimal foliated by circles in parallel planes, then these planes are orthogonal to the vector $(\alpha,\beta,\gamma)$ and the surface must be rotational. We also classify all minimal surfaces of translation type

Special Weingarten surfaces foliated by circles

In this paper we study surfaces in Euclidean 3-space foliated by pieces of circles and that satisfy a Weingarten condition of type $a H+b K=c$, where $a,b$ and $c$ are constant and $H$ and $K$ denote the mean curvature and the Gauss curvature respectively. We prove that a such surface must be a surface of revolution, a Riemann minimal surface or a generalized cone.Comment: 17 page

Parabolic surfaces in hyperbolic space with constant curvature

We study parabolic linear Weingarten surfaces in hyperbolic space \rlopezh^3. In particular, we classify two family of parabolic surfaces: surfaces with constant Gaussian curvature and surfaces that satisfy the relation $a\kappa_1+b\kappa_2=c$, where $\kappa_i$ are the principal curvatures, and $a,b$ and $c$ are constant.Comment: 8 pages, 6 figures. This is the text of the talk presented in the International Congress on Pure and Applied Differential Geometry, PADGE 2007, 10-13 April, 2007. This is a brief of two preprints due to the author and that appear in the References sectio

A new proof of a characterization of small spherical caps

It is known that planar disks and small spherical caps are the only constant mean curvature graphs whose boundary is a round circle. Usually, the proof invokes the Maximum Principle for elliptic equations. This paper presents a new proof of this result motivated by an article due to Reilly. Our proof utilizes a flux formula for surfaces with constant mean curvature together with integral equalities on the surface.Comment: 11 pages. to appear in Results in Mathematic

Parabolic Weingarten surfaces in hyperbolic space

A surface in hyperbolic space \h^3 invariant by a group of parabolic isometries is called a parabolic surface. In this paper we investigate parabolic surfaces of \h^3 that satisfy a linear Weingarten relation of the form $a\kappa_1+b\kappa_2=c$ or $aH+bK=c$, where a,b,c\in \r and, as usual, $\kappa_i$ are the principal curvatures, $H$ is the mean curvature and $K$ is de Gaussian curvature. We classify all parabolic linear Weingarten surfaces in hyperbolic space.Comment: 22 pages, 10 figures; This work was announced in arXiv:0704.275

The Lorentzian version of a theorem of Krust

In Lorentz-Minkowski space, we prove that the conjugate surface of a maximal graph over a convex domain is also a graph. We provide three proofs of this result that show a suitable correspondence between maximal surfaces in Lorentz-Minkowski space and minimal surfaces in Euclidean space

Capillary channels in a gravitational field

The liquid shape between two vertical parallel plates in a gravity field due to capillary forces is studied. When the physical system achieves its mechanical equilibrium, the capillary surface has mean curvature proportional to its height above a horizontal reference plane and it meets the vertical walls in a prescribed angle. We examine the shapes of these interfaces and their qualitative properties depending on the sign of the capillary constant. We focus to obtain estimates of the size of the meniscus, as for example, its height and volume.Comment: 28 page