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Rafael López

Publications and source records attributed to Rafael López.

At least 37 records · Page 2Linked to original sources
arXiv2025

Two classification results for stationary surfaces of the least moment of inertia

A surface in Euclidean space $\r^3$ is said to be an $α$-stationary surface if it is a critical point of the energy $\int_Σ|p|^α$, where $α\in\r$. We prove that all ruled $α$-stationary surfaces are vector planes (for all $α$) and a type of elongated helicoids (for $α=1$). The second result of classification asserts that if $α\not=-2,-4$, any $α$-stationary surface foliated by circles must be a rotational surface. If $α=-4$, the surface is the inversion of a plane, a helicoid, a catenoid or an Riemann minimal example. If $α=-2$, we find many non-spherical cyclic $(-2)$-stationary surfaces.

math.DG↗
arXiv2025

Extension of a problem of Euler in Lorentz-Minkowski plane

In this paper we study curves in Lorentz-Minkowski space $\mathbb{L}^2$ that are critical points of the moment of inertia with respect to the origin. This extends a problem posed by Euler in the Lorentzian setting. We obtain explicit solutions for stationary curves in $\mathbb{L}^2$ distinguishing if the curve is spacelike or timelike. We also give a method to carry stationary spacelike curves into stationary timelike curves and vice versa via symmetries and inversions about the lightlike cone. Finally, we solve the problem of maximizing the energy among all spacelike curves joining two given points which are collinear with the origin.

math.DG↗
arXiv2025

Classification of ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia

In this paper, we study hypersurfaces in Lorentz-Minkowski space $\mathbb{L}^{n+1}$ that are stationary for the moment of inertia with respect to the origin. After giving examples and applications of the maximum principle, we classify, in $\mathbb{L}^3$, all stationary surfaces that are ruled surfaces. We prove that planes are the only cylindrical stationary surfaces. If the surface is not cylindrical, the classification depends on the causal character of the rulings. In addition, we provide explicit parametrizations of all ruled stationary surfaces.

math.DG↗
arXiv2025

Invariant $λ$-translators for the Gauss curvature flow in Euclidean space

A $λ$-translator is a surface in Euclidean space $\mathbb{R}^3$ whose Gauss curvature $K$ satisfies $K=\langle N, \vec{v} \rangle +λ$, where $N$ is the Gauss map, $\vec{v}$ is a fixed direction, and $λ\in \mathbb{R}$. In this paper, we classify all $λ$-translators that are invariant by a one-parameter group of translations and a one-parameter group of rotations.

math.DG↗
arXiv2025

Singular minimal surfaces with constant curvature

We prove that singular minimal surfaces with constant Gauss curvature are planes, spheres and cylindrical surfaces. We also classify all singular minimal surfaces with a constant principal curvature and singular minimal surfaces with constant mean curvature.

math.DG↗
arXiv2025

A Note on Helicoidal Singular Minimal Surfaces

Let $α\in\r$ and let $\vec{v}\in\r^3$ be a unit vector. A singular minimal surface $Σ$ in Euclidean space is a surface $Σ$ whose mean curvature $H$ satisfies $H=α\frac{\langle N,\vec{v}\rangle}{\langle p,\vec{v}\rangle}$, where $N$ is the unit normal vector of $Σ$. In this short note we study singular minimal surfaces which are invariant by a one-parameter group of helicoidal motions. We prove that if $Σ$ is a helicoidal singular minimal surface, then the axis of the helicoidal motion is orthogonal to $\vec{v}$, $α=-1$ and $Σ$ is a circular right cylinder.

math.DG↗
arXiv2025

Maximal translation surfaces in Lorentz-Minkowski space

A translation surface in Lorentz-Minkowski space $\rr^3$ is a surface defined as the sum of two spatial curves. In this paper we present a classification of maximal surfaces of translation type. We prove that if a generating curve is planar, then the other generating curve is also planar. We give a full description of these surfaces. In case that both curves are of Frenet type, we generalize the Scherk surfaces. In case that a curve is a pseudo-null curve, we obtain new examples of maximal surfaces which have not counterparts in Euclidean space.

math.DG↗
arXiv2025

A note on helices in the Euclidean space and in the hyperbolic space

We prove that general helices in Euclidean space for Killing vector fields associated to rotations are helices, that is, curves with constant curvature and constant torsion. In hyperbolic space $\h^3$, we obtain the parametrization of helices for the Killing vector fields associated to hyperbolic rotations, spherical rotations and parabolic rotations. It is proved that these helices are geodesics in suitable surfaces of $\h^3$.

math.DG↗
arXiv2025

Rigidity of solitons of the Gauss curvature flow in Euclidean space

In this paper, we consider $λ$-translating solitons and $λ$-shrinkers of the Gauss curvature flow in Euclidean space. We prove that planes and circular cylinders are the only $λ$-translating solitons with constant mean curvature. We also prove that planes, spheres and circular cylinders are the only $λ$-shrinkers with constant mean curvature. We give a classification of the $λ$-translating solitons and $λ$-shrinkers with one constant principal curvature.

math.DG↗
arXiv2025

Stationary surfaces for the moment of inertia with constant Gauss curvature

Consider the energy $E_α[Σ]=\int_Σ|p|^α\, dΣ$, where $Σ$ is a surface in Euclidean space $\r^3$ and $α\in\r$. We prove that planes and spheres are the only stationary surfaces for $E_α$ with constant Gauss curvature. We also characterize these surfaces assuming that a principal curvature is constant or that the mean curvature is constant.

math.DG↗
arXiv2025

A connection between minimal surfaces and the two-dimensional analogues of a problem of Euler

If $α\in\r$, an $α$-stationary surface in Euclidean space is a surface $Σ$ whose mean curvature $H$ satisfies $H(p)=α|p|^{-2} \langleν,p\rangle$, $p\inΣ$. These surfaces generalize in dimension two a classical family of curves studied by Euler which are critical points of the moment of inertia of planar curves. In this paper we establish, via inversions, a one-to-one correspondence between $α$-stationary surfaces and $-(α+4)$-stationary surfaces. In particular, there is a correspondence between $-4$-stationary surfaces and minimal surfaces. Using this duality we give some results of uniqueness of $-4$-stationary surfaces and we solve the Börling problem.

math.DG↗
arXiv2025

Axisymmetric stationary surfaces for the moment of inertia

We investigate axisymmetric surfaces in Euclidean space that are stationary for the energy $E_α=\int_Σ|p|^α\, dΣ$. By using a phase plane analysis, we classify these surfaces when they intersect orthogonally the rotation axis. We also give some applications of the maximum principle characterizing the closed stationary surfaces and the compact stationary surfaces with boundary a circle when $α=-2$. Finally, we prove that helicoidal stationary surfaces must be rotational surfaces.

math.DG↗
arXiv2025

Reduced SIGMA Basis Sets: a new family of SIGMA basis sets for molecular calculations

A new family of Gaussian-type SIGMA basis sets, termed reduced SIGMA basis sets, is introduced and preliminarily tested. Sharing the same composition as Dunning basis sets, they enhance performance by reducing linear dependencies in large systems, thereby improving convergence and lowering computational costs for such systems.

physics.comp-ph↗
arXiv2025

Capillary liquid channels in cylindrical support surfaces: stability and bifurcation

Planes and circular cylinders are models of interfaces of a fluid when the support surface is translationally invariant in a direction of the space. After a study of the eigenvalues of the Jacobi operator, it is investigated when planar strips and sections of circular cylinders are stable in cylindrical symmetric support surfaces. This analysis depends on the curvature of the support at the contact points with the interface. The Plateau-Rayleigh instability phenomenon is studied finding the critical value $h_0>0$ such that rectangular pieces of planar strips or circular cylinders of length greater than $h_0$ are necessarily unstable. It is also studied when new morphologies of capillary surfaces can emerge from given circular cylinders. Using the method of bifurcation by simple eigenvalues, we establish conditions on the support surface that prove that when $0$ is a simple eigenvalue of the Jacobi operator, there is bifurcation from explicit circular cylinders. It will be presented examples of supports (parabolic and catenary cylinders) where this bifurcation appears.

math.DG↗
arXiv2025

Stability of vertical and radial graphs in the Euclidean space with density

It is proved that vertical graphs and radial graphs are strongly stable for a certain type of densities in Euclidean space ${\mathbb R}^{n+1}$. Particular cases of these densities include translators, expanders and singular minimal hypersurfaces. Using techniques of calibrations, it is also proved that for densities depending on a spatial coordinate, stationary vertical graphs are weighted minimizers in a certain class of hypersurfaces.

math.DG↗
arXiv2024

Differential geometry of curves in dual space

We introduce the Frenet theory of curves in dual space $\d^3$. After defining the curvature and the torsion of a curve, we classify all curves in dual plane with constant curvature. We also establish the fundamental theorem of existence in the theory of dual curves, proving that there is a dual curve with prescribed curvature and torsion. Finally we classify all dual curves with constant curvature and torsion.

math.DG↗
arXiv2024

On intrinsic rotational surfaces in the Lorentz-Minkowski space

Spacelike intrinsic rotational surfaces with constant mean curvature in the Lorentz-Minkowski space $\E_1^3$ have been recently investigated by Brander et al., extending the known Smyth's surfaces in Euclidean space. Assuming that the surface is intrinsic rotational with coordinates $(u,v)$ and conformal factor $ρ(u)^2$, we replace the constancy of the mean curvature with the property that the Weingarten endomorphism $A$ can be expressed as $Φ_{-α(v)}\left(\begin{array}{ll}λ_1(u)&0\\ 0&λ_2(u)\end{array}\right)Φ_{α(v)}$, where $Φ_{α(v)}$ is the (Euclidean or hyperbolic) rotation of angle $α(v)$ at each tangent plane and $λ_i$ are the principal curvatures. Under these conditions, it is proved that the mean curvature is constant and $α$ is a linear function. This result also covers the case that the surface is timelike. If the mean curvature is zero, we determine all spacelike and timelike intrinsic rotational surfaces with rotational angle $α$. This family of surfaces includes the spacelike and timelike Enneper surfaces.

math.DG↗