arXiv subjects
Publications and source records attributed to Rafael López.
The generating curves of rotational minimal surfaces in the de Sitter space $\s_1^3$ are characterized as solutions of a variational problem. It is proved that these curves are the critical points of the center of mass among all curves of $\s_1^2$ with prescribed endpoints and fixed length. This extends the known properties of the catenary and the catenoid in the Euclidean setting.
A new family of Gaussian-type basis sets named sigma basis sets is presented and preliminarily tested. Sigma basis sets for H, C, N, O and P are reported and their performance is tested in some atomic and molecular calculations.
A spacelike surface in Minkowski space $\mathbb{R}_1^3$ is called a $K^α$-translator of the flow by the powers of Gauss curvature if satisfies $K^α= \langle N,\vec{v}\rangle$, $α\neq 0$, where $K$ is the Gauss curvature, $N$ is the unit normal vector field and $\vec{v}$ is a direction of $\mathbb{R}_1^3$. In this paper, we classify all rotational $K^α$-translators. This classification will depend on the causal character of the rotation axis. Although the theory of the $K^α$-flow holds for spacelike surfaces, the equation describing $K^α$-translators is still valid for timelike surfaces. So we also investigate the timelike rotational surfaces that satisfy the same prescribing Gauss curvature equation.
In this paper we study the linear Weingarten equation defined by the fully non-linear PDE $$a\, \mbox{div}\frac{Du}{\sqrt{1+|Du|^2}}+b\, \frac{\mbox{det}D^2u}{(1+|Du|^2)^2}=ϕ\left(\frac{1}{\sqrt{1+|Du|^2}}\right)$$ in a domain $Ω\subset\mathbb{R}^2$, where $ϕ\in C^1([-1,1])$ and $a,b\in\mathbb{R}$. We approach the existence of radial solutions when $Ω$ is a disk of small radius, giving an affirmative answer when the PDE is of elliptic type. In the hyperbolic case we show that no radial solution exists, while in the parabolic case we find explicitly all the solutions. Finally, in the elliptic case we prove uniqueness and symmetry results concerning the Dirichlet problem of such equation.
A $K^α$-translator is a surface in Euclidean space $\r^3$ that moves by translations in a spatial direction and under the $K^α$-flow, where $K$ is the Gauss curvature and $α$ is a constant. We classify all $K^α$-translators that are rotationally symmetric. In particular, we prove that for each $α$ there is a $K^α$-translator intersecting orthogonally the rotation axis. We also describe all $K^α$-translators invariant by a uniparametric group of helicoidal motions and the translators obtained by separation of variables.
We introduce a new class of surfaces in Euclidean $3$-space, called surfaces of osculating circles, using the concept of osculating circle of a regular curve. These surfaces contain a uniparametric family of planar lines of curvature. In this paper, we classify those ones that are canal surfaces and Weingarten surfaces.
Riemann zero mean curvature examples in the Lorentz-Minkowski space are surfaces with zero mean curvature foliated by circles contained in parallel planes. In contrast to the Euclidean case, this family of surfaces presents new and rich features because of the variety of types of circles. In this paper, we give a geometric description of these examples when the circles are contained in spacelike planes and timelike planes.
We classify cylindrical surfaces in the Euclidean space whose mean curvature is a $n$th-power of the distance to a reference plane. The generating curves of these surfaces, called $n$-elastic curves, have a variational characterization as critical points of a curvature energy generalizing the classical elastic energy. We give a full description of such curves obtaining, in some particular cases, closed curves including simple ones.
This article examines the shape of a surface obtained by a hanging flexible, inelastic material with prescribed area and boundary curve. The shape of this surface, after being turned upside down, is a model for cupolas (or domes) under the simple hypothesis of compression. Investigating the rotational examples, we provide and illustrate a novel design for a roof which has the extraordinary property that its shape, although natural, is modeled by a surface of revolution whose axis of rotation is horizontal.
We investigate the problem of determining the planar curves that describe ramps where a particle of mass $m$ moves with constant-speed when is subject to the action of the friction force and a force whose magnitude $F(r)$ depends only on the distance $r$ from the origin. In this paper we describe all the constant-speed ramps for the case $F(r)=-m/r$. We show the circles and the logarithmic spirals play an important role. No only they are solutions but every other solution approaches either a circle or a logarithmic spiral.
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.
In Euclidean space, we investigate surfaces whose mean curvature $H$ satisfies the equation $H=α\langle N,\mathbf{x}\rangle+λ$, where $N$ is the Gauss map, $\mathbf{x}$ is the position vector and $α$ and $λ$ are two constants. There surfaces generalize self-shrinkers and self-expanders of the mean curvature flow. We classify the ruled surfaces and the translation surfaces, proving that they are cylindrical surfaces.
Consider the Lorentz-Minkowski $3$-space $\mathbb{L}^3$ with the metric $dx^2+dy^2-dz^2$ in canonical coordinates $(x,y,z)$. A surface in $\mathbb{L}^3$ is said to be separable if satisfies an equation of the form $f(x)+g(y)+h(z)=0$ for some smooth functions $f$, $g$ and $h$ defined in open intervals of the real line. In this article we classify all zero mean curvature surfaces of separable type, providing a method of construction of examples.
In this paper, we classify the rotational surfaces with constant skew curvature in $3$-space forms. We also give a variational characterization of the profile curves of these surfaces as critical points of a curvature energy involving the exponential of the curvature of the curve. Finally, we provide a converse process to produce all rotational surfaces with constant skew curvature based on the evolution of a critical curve under the flow of its binormal vector field with prescribed velocity.
A surface in a Riemannian space is called of constant astigmatism if the difference between the principal radii of curvatures at each point is a constant function. In this paper we give a classification of all rotational surfaces of constant astigmatism in space forms. We also prove that the generating curves of such surfaces are critical points of a variational problem for a curvature energy. Using the description of these curves, we locally construct all rotational surfaces of constant astigmatism as the associated binormal evolution surfaces from the generating curves.
In this paper we generalize in Lorentz-Minkowski space $ł^3$ the two-dimensional analogue of the catenary of Euclidean space. We solve the Dirichlet problem for bounded mean convex domains and spacelike boundary data that have a spacelike extension to the domain. We also classify all singular maximal surfaces of $ł^3$ invariant by a uniparametric group of translations and rotations.
In this paper we are dealing with two classes of mean curvature type problems that generalize the translating soliton problem. A first result proves that the solutions to these problems have unique interior critical points. Using this uniqueness result, we next derive a priori $C^0$ and $C^1$ estimates for the solutions to these problems, by means of some minimum principles for appropriate $P$-functions, in the sense of L.E. Payne.
We establish gradient estimates for solutions to the Dirichlet problem for the constant mean curvature equation in hyperbolic space. We obtain these estimates on bounded strictly convex domains by using the maximum principles theory of $Φ$-functions of Payne and Philippin. These estimates are then employed to solve the Dirichlet problem when the mean curvature $H$ satisfies $H<1$ under suitable boundary conditions.