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Rafael López

Publications and source records attributed to Rafael López.

At least 91 records · Page 5Linked to original sources
arXiv2019

The Dirichlet problem of the constant mean curvature equation in Lorentz-Minkowski space and in Euclidean space

We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the spacelike condition in the Lorentz-Minkowski space allows to drop the hypothesis on the mean convexity which is required in the Euclidean case.

math.DG↗
arXiv2019

A characteristic property of Delaunay surfaces

We prove that Delaunay surfaces, except the plane and the catenoid, are the only surfaces in Euclidean space with nonzero constant mean curvature that can be expressed as an implicit equation of type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are smooth real functions of one variable.

math.DG↗
arXiv2019

Classification of separable surfaces with constant Gaussian curvature

We classify all surfaces with constant Gaussian curvature $K$ in Euclidean $3$-space that can be expressed as an implicit equation of type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are real functions of one variable. If $K=0$, we prove that the surface is a surface of revolution, a cylindrical surface or a conical surface, obtaining explicit parametrizations of such surfaces. If $K\not=0$, we prove that the surface is a surface of revolution.

math.DG↗
arXiv2018

The translating soliton equation

We give an analytic approach to the translating soliton equation with a special emphasis in the study of the Dirichlet problem in convex domains of the plane.

math.DG↗
arXiv2018

Compact singular minimal surfaces with boundary

We study of the shape of a compact singular minimal surface in terms of the geometry of its boundary, asking what type of {\it a priori} information can be obtained on the surface from the knowledge of its boundary. We derive estimates of the area and the height in terms of the boundary. In case that the boundary is a circle, we study under what conditions the surface is rotational. Finally, we deduce non-existence results when the boundary is formed by two curves that are sufficiently far apart.

math.DG↗
arXiv2018

Classification and construction of minimal translation surfaces in Euclidean space

A translation surface of Euclidean space $\r^3$ is the sum of two regular curves $α$ and $β$, called the generating curves. In this paper we classify the minimal translation surfaces of $\r^3$ and we give a method of construction of explicit examples. Besides the plane and the minimal surfaces of Scherk type, it is proved that up to reparameterizations of the generating curves, any minimal translation surface is described as $Ψ(s,t)=α(s)+α(t)$, where $α$ is a curve parameterized by arc length $s$, its curvature $κ$ is a positive solution of the autonomous ODE $(y')^2+y^4+c_3y^2+c_1^2y^{-2}+c_1c_2=0$ and its torsion is $τ(s)=c_1/κ(s)^2$. Here $c_1\not=0$, $c_2$ and $c_3$ are constants such that the cubic equation $-λ^3+c_2λ^2-c_3λ+c_1=0$ has three real roots $λ_1$, $λ_2$ and $λ_3$.

math.DG↗
arXiv2018

Classification of rotational surfaces in Euclidean space satisfying a linear relation between their principal curvatures

We classify all rotational surfaces in Euclidean space whose principal curvatures $κ_1$ and $κ_2$ satisfy the linear relation $κ_1=aκ_2+b$, where $a$ and $b$ are two constants. We give a variational characterization of these surfaces in terms of its generating curve. As a consequence of our classification, we find closed (embedded and not embedded) surfaces and periodic (embedded and not embedded) surfaces with a geometric behaviour similar to Delaunay surfaces.

math.DG↗
arXiv2018

The Dirichlet problem for the $α$-translating soliton equation on a strip

We prove the existence of classical solutions to the Dirichlet problem for the $α$-translating soliton equation defined in a strip of $\r^2$. We use the Perron method where a family of grim reapers are employed as barriers for solving the Dirichlet problem when the boundary data is formed by two copies of a convex function.

math.DG↗
arXiv2018

Invariant surfaces in Euclidean space with a log-linear density

A $λ$-translating soliton with density vector $\vec{v}$ is a surface in Euclidean space whose mean curvature $H$ satisfies $2H=2λ+\langle N,\vec{v}\rangle$, where $N$ is the Gauss map. We classify all $λ$-translating solitons that are invariant by a one-parameter group of translations and a one-parameter group of rotations.

math.DG↗
arXiv2018

Compact $λ$-translating solitons with boundary

A $λ$-translating soliton with density vector $\vec{v}$ is a surface $Σ$ in Euclidean space ${\mathbb R}^3$ whose mean curvature $H$ satisfies $2H=2λ+\langle N,\vec{v}\rangle$, where $N$ is the Gauss map of $Σ$. In this article we study the shape of a compact $λ$-translating soliton in terms of its boundary. If $Γ$ is a given closed curve, we deduce under what conditions on $λ$ there exists a compact $λ$-translating soliton $Σ$ with boundary $Γ$ and we provide estimates of the surface area in relation with the height of $Σ$. Finally we study the shape of $Σ$ related with the one of $Γ$, in particular, we give conditions that assert that $Σ$ inherits the symmetries of its boundary $Γ$.

math.DG↗
arXiv2017

On the duality between rotational minimal surfaces and maximal surfaces

We investigate the duality between minimal surfaces in Euclidean space and maximal surfaces in Lorentz-Minkowski space in the family of rotational surfaces. We study if the dual surfaces of two congruent rotational minimal (or maximal) surfaces are congruent. We show that in the duality process by means of a one-parameter group of rotations, it appears the family of Bonnet minimal (maximal) surfaces and the Goursat transformations.

math.DG↗
arXiv2016

Explicit Björling Surfaces with Prescribed Geometry

We develop a new method to construct explicit, regular minimal surfaces in Euclidean space that are defined on the entire complex plane with controlled geometry. More precisely we show that for a large class of planar curves $(x(t), y(t))$ one can find a third coordinate $z(t)$ and normal fields $n(t)$ along the space curve $c(t)=(x(t), y(t), z(t))$ so that the Björling formula applied to $c(t)$ and $n(t)$ can be explicitly evaluated. We give many examples.

math.DG↗
arXiv2014

Capillary surfaces of constant mean curvature in a right solid cylinder

In this paper we investigate constant mean curvature surfaces with nonempty boundary in Euclidean space that meet a right cylinder at a constant angle along the boundary. If the surface lies inside of the cylinder, we obtain some results of symmetry by using the Alexandrov reflection method. When the mean curvature is zero, we give sufficient conditions to obtain that the surface is part of a plane or a catenoid.

math.DG↗
arXiv2014

Constant mean curvature surfaces with boundary on a sphere

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle. We study under what geometric conditions the surface must be spherical. Our results apply in many scenarios in physics where in absence of gravity a liquid drop is deposited on a round solid ball and the air-liquid interface is a critical point for area under all variations that preserve the enclosed volume.

math.DG↗