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Rafael López

Publications and source records attributed to Rafael López.

At least 109 records · Page 6Linked to original sources
arXiv2014

Capillary surfaces in a cone

We show that a capillary surface in a solid cone, that is, a surface that has constant mean curvature and the boundary of surface meets the boundary of the cone with a constant angle, is radially graphical if the mean curvature is non-positive with respect to the Gauss map pointing toward the domain bounded by the surface and the boundary of the cone. In the particular case that the cone is circular, we prove that the surface is a spherical cap or a planar disc. The proofs are based on an extension of the Alexandrov reflection method by using inversions about spheres.

math.DG↗
arXiv2014

Translation surfaces of linear Weingarten type

We give a relatively simple proof that a translation surface in Euclidean space that satisfies a relation of type $aH+bK=c$, for some real numbers $a,b,c$, where $H$ and $K$ are the mean curvature and the Gauss curvature of the surface, respectively, must have $a=0$ or $b=0$, and thus, $K$ is constant or $H$ is constant. Our method of proof extends to the Lorentzian ambient space.

math.DG↗
arXiv2014

Surfaces in Sol$_3$ space foliated by circles

In this paper we study surfaces foliated by a uniparametric family of circles in the homogeneous space Sol$_3$. We prove that there do not exist such surfaces with zero mean curvature or with zero Gaussian curvature. We extend this study considering surfaces foliated by geodesics, equidistant lines or horocycles in totally geodesic planes and we classify all such surfaces under the assumption of minimality or flatness.

math.DG↗
arXiv2014

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 $ϕ(x,y,z)=αx+βy+γy$, where $α,β,γ$ are real numbers not all zero. We prove that if a surface is $ϕ$-minimal foliated by circles in parallel planes, then these planes are orthogonal to the vector $(α,β,γ)$ and the surface must be rotational. We also classify all minimal surfaces of translation type.

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arXiv2011

Spacelike surfaces with free boundary in the Lorentz-Minkowski space

We investigate a variational problem in the Lorentz-Minkowski space $ł^3$ whose critical points are spacelike surfaces with constant mean curvature and making constant contact angle with a given support surface along its common boundary. We show that if the support surface is a pseudosphere, then the surface is a planar disc or a hyperbolic cap. We also study the problem of spacelike hypersurfaces with free boundary in the higher dimensional Lorentz-Minkowski space $ł^{n+1}$.

math.DG↗
arXiv2011

Bifurcation of cylinders for wetting and dewetting models with striped geometry

We show that some pieces of cylinders bounded by two parallel straight-lines bifurcate in a family of periodic non-rotational surfaces with constant mean curvature and with the same boundary conditions. These cylinders are initial interfaces in a problem of microscale range modeling the morphologies that adopt a liquid deposited in a chemically structured substrate with striped geometry or a liquid contained in a right wedge with Dirichlet and capillary boundary condition on the edges of the wedge. Experiments show that starting from these cylinders and once reached a certain stage, the shape of the liquid changes drastically in an abrupt manner. Studying the stability of such cylinders, the paper provides a mathematical proof of the existence of these new interfaces obtained in experiments. The analysis is based on the theory of bifurcation by simple eigenvalues of Crandall-Rabinowitz.

math.DG↗
arXiv2011

Invariant surfaces in Sol$_3$ with constant mean curvature and their computer graphics

In Sol$_3$ space there are three uniparametric groups of isometries. In this work we study constant mean curvature surfaces invariant by one of these groups. We analyze the geometric properties of these surfaces by means of their computer graphics. We construct explicit examples of minimal surfaces and we shall relate them with recent examples of spheres with constant mean curvature.

math.DG↗
arXiv2010

Minimal translation surfaces in $Sol_3$

In the homogeneous space Sol$_3$, a translation surface is parameterized by $x(s,t)=α(s)\astβ(t)$, where $α$ and $β$ are curves contained in coordinate planes and $\ast$ denotes the group operation of Sol$_3$. In this paper we study translation surfaces in Sol$_3$ whose mean curvature vanishes.

math.DG↗
arXiv2010

Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

In this work we find all helicoidal surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is $0$ or $1$ and that the surface is ruled. If the generating curve is a Lorentzian circle, we show that the only possibility is that the axis is spacelike and the center of the circle lies in the axis.

math.DG↗
arXiv2010

Spacelike surfaces in Minkowski space satisfying a linear relation between their principal curvatures

In this work, we consider spacelike surfaces in Minkowski space $\hbox{\bf E}%_{1}^{3}$ that satisfy a linear Weingarten condition of type $κ_{1}=mκ_{2}+n$, where $m$ and $n$ are constant and $κ_{1}$ and $κ_{2}$ denote the principal curvatures at each point of the surface. We study the family of surfaces foliated by a uniparametric family of circles in parallel planes. We prove that the surface must be rotational or the surface is part of the family of Riemann examples of maximal surfaces ($m=-1$, $n=0$). Finally, we consider the class of rotational surfaces for the case $n=0$, obtaining a first integration if the axis is timelike and spacelike and a complete description if the axis is lightlike.

math.DG↗
arXiv2010

Non-degenerate surfaces of revolution in Minkowski space that satisfy the relation $aH+bK=c$

In this work, we study spacelike and timelike surfaces of revolution in Minkowski space $\e_{1}^{3}$ that satisfy $aH+bK=c$, where $H$ and $K$ denote the mean curvature and the Gauss curvature of the surface and $a$, $b$ and $c$ are constants. The classification depends on the causal character of the axis of revolution and in all the cases, we obtain a first integral of the equation of the generating curve of the surface.

math.DG↗
arXiv2009

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.

math.DG↗
arXiv2009

Invariant surfaces in homogenous space Sol with constant curvature

A surface in homogenous space Sol is said to be an invariant surface if it is invariant under some of the two 1-parameter groups of isometries of the ambient space whose fix point sets are totally geodesic surfaces. In this work we study invariant surfaces that satisfy a certain condition on their curvatures. We classify invariant surfaces with constant mean curvature and constant Gaussian curvature. Also, we characterize invariant surfaces that satisfy a linear Weingarten relation.

math.DG↗
arXiv2009

Linear Weingarten surfaces foliated by circles in Minkowski space

In this work, we study spacelike surfaces in Minkowski space $E_1^3$ foliated by pieces of circles and that satisfy a linear 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 show that such surfaces must be surfaces of revolution or surfaces with constant mean curvature H=0 or surfaces with constant Gauss curvature K=0.

math.DG↗
arXiv2009

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.

math.DG↗