arXiv subjects
Publications and source records attributed to Rafael López.
We prove that a compact immersed disk of constant mean curvature in $\mathbb S^3$ or in $\mathbb H^3$ whose boundary is mapped diffeomorphically onto a circle is an embedded totally umbilical disk.
In this paper, we classify all solitons of the Gauss curvature flow in the three-dimensional Heisenberg group $\mathrm{Nil}_3$ that are invariant under a one-parameter group of ambient isometries. By means of the four canonical types of Killing vector fields and the three families of invariant surfaces (vertical translations, horizontal translations, and helicoidal motions), we analyze the twelve resulting types of possible solitons. In some cases, there do not exist any invariant solitons; in others, we find explicit parametrizations, or describe their geometric properties.
We investigate the geometry of plane curves in $\mathbb{R}^2$ endowed with a canonical linear connection. By introducing the notions of the tangential and geodesic curvatures, we prove the existence and uniqueness theorems for curves with prescribed curvatures. We find all curves with constant curvature and we generalize the curves whose curvatures are linear functions of the arc length parameter.
We extend Newton's problem of minimal resistance to Riemannian surfaces endowed with a geodesic coordinate system, which includes the two-dimensional space forms such as the sphere and the hyperbolic plane. Assuming that the fluid particles flow along radial geodesics, we derive the resistance functional and prove that its smooth extremals are the loxodromes of the surface. Furthermore, we analyze the constrained minimization problem, establishing the absence of strong local minima for smooth extremals, and characterizing their global minimizers.
We extend Newton's problem of minimal resistance to the Lorentz-Minkowski space. We derive the functional energy and determine the Euler-Lagrange equation. In contrast to the Euclidean case, this equation is quasilinear elliptic, and thus, a maximum principle holds in this context. We obtain the solutions of separable variables of this equation via separation of variables. Furthermore, we find all radial solutions to the problem, which present conical singularities at the origin. We also analyze the Single Shock Condition.
We classify invariant surfaces in the 3-dimensional solvable Lie group $\sol$ that act as solitons for the Gauss curvature flow. We consider solitons associated with the canonical basis of Killing vector fields $\{F_1, F_2, F_3\}$, where $F_1$ and $F_2$ generate horizontal translations and $F_3$ generates the scaling isometry. We establish rigidity results for $F_3$-invariant surfaces, proving that specific totally geodesic vertical planes are the only $F_1$- and $F_2$-solitons. For $F_1$-invariant surfaces, we establish the main geometric properties of $F_2$- and $F_3$-solitons in both the extrinsic and intrinsic Gauss curvature.
This paper investigates the nonlinear dynamics of Newton's problem of minimal resistance in radial fields. We move beyond classical translational symmetry to analyze two non-equilibrium scenarios: a scale-invariant free expansion and an incompressible source flow. Our analysis reveals that the scale-invariant model suffers from a symmetry-breaking instability (loss of ellipticity) that necessitates geometric truncation. Conversely, we prove that the incompressible flow acts as a structural regularizer, admitting unique, smooth, and strictly concave solutions. These findings provide new qualitative insights into how physical conservation laws ensure the regularity and symmetry of optimal configurations in high-speed central flows, bridging the gap between variational calculus and the physics of complex systems.
We classify all ruled surfaces in Euclidean space that are critical points of the Dirichlet energy, obtaining explicit parametrizations of these surfaces.
We study the inverse curve shortening flow in the hyperbolic plane $\h^2$. We classify all solitons with respect to parabolic and conformal vector fields of $\h^2$. In the upper half-plane model of $\h^2$, we prove that parabolic solitons are all graphs on the $y$-axis, whereas conformal solitons are graphs on the $x$-axis. We study the concavity of these solitons and when they approach the coordinate axes.
We establish necessary conditions for a regular curve to lie on a circular cylinder in terms of its curvature $κ$ and torsion $τ$. By identifying a fundamental function $ψ= \sin^2 α$, representing the squared sine of the angle between the tangent vector and the axis of the cylinder, we reduce the geometric inclusion problem to a compatibility condition between an explicit eighth-degree polynomial equation and a differential equation for $ψ$. This approach yields a single ODE involving only $κ$ and $τ$ that governs the inclusion of the curve in the cylinder. The robustness of this framework is demonstrated through specific examples of cylindrical curves. Furthermore, we analyze the case of curves with constant curvature $κ_0$, obtaining an explicit ODE for the torsion. Remarkably, we prove that if $κ_0 = 1/ρ$, this equation admits an explicit, exact solution for $τ$.
We define a new notion of translations in the hyperbolic plane and explicitly solve the equation of the curve shortening flow. Next, we consider the class of ancient convex solutions and solve the equation of the curve shortening flow when the curvature function is given by separation of variables. Lastly, we prove some area estimates for closed ancient solutions of the curve shortening flow.
In this paper, we study surfaces $z=φ(x,y)$ in Euclidean space that satisfy the equation $φ_{xx}+φ_{yy}=\fracΛ{2}$ where $Λ\in\r$ is a real constant. We classify these surfaces when they are the zero level sets of an implicit equation of the type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are smooth functions of one variable. If $Λ=0$, we find a large family of surfaces with interesting symmetry properties. However, if $Λ\not=0$, we show that the surfaces must be either surfaces of revolution or of the type $z=f(x)+g(y)$; furthermore, explicit parametrizations of these surfaces are obtained.
This paper investigates the Newton's problem of minimal resistance for a body moving through a fluid whose density decreases exponentially with altitude. We prove the local existence and regularity of radial solutions $u(r)$ satisfying the initial conditions $u(0)=u'(0)=0$ using a fixed-point theorem. We show that the maximal domain of the solution is finite, $[0, r_M)$, terminating at a critical slope $u'(r_M) = \frac{1}{\sqrt{3}}$.
In Euclidean space we study surfaces with constant anisotropic mean curvature $Λ$ of the Dirichlet energy $\int_Ω( |Du|^2+Λu)$. We prove the existence of non-rotational surfaces with $Λ=0$ and foliated by a one-parameter family of circles contained in horizontal planes obtaining a geometric description of them. These surfaces extend the known Riemann examples of the theory of minimal surfaces to the anisotropic context of the Dirichlet energy. More general, we classify all surfaces with zero anisotropic mean curvature foliated by circles proving that either the surface is axially symmetric about the $z$-axis or the surface belongs to one of the above examples. We also study the case that the anisotropic mean curvature is a non-zero constant.
We classify all homothetical surfaces with constant mean curvature $H$ in the hyperbolic space $\mathbb{H}^3$. Using the upper half-space model with standard coordinates $(x,y,z)$, these surfaces are defined by the relation $z = ϕ(x)ψ(y)$, where $ϕ$ and $ψ$ are smooth functions of one variable. We demonstrate that any such surface is necessarily parabolic, meaning that either $ϕ$ or $ψ$ is a constant function. Our results cover the minimal case ($H=0$), the case $H^2 \neq 1$, and the critical case $H^2=1$, thereby extending the existing classification of parabolic surfaces in hyperbolic space.
In this paper, we study a dual analogue of the classical catenary within the class of admissible curves in the dual plane $\mathbb{D}^2$. We introduce $α$-catenaries in $\mathbb{D}^2$ as stationary points of a potential energy functional, where $α\in \mathbb{R}$ is a real parameter. We derive the corresponding Euler-Lagrange equations and obtain explicit equations of these curves for specific values of $α$. Furthermore, we establish a geometric characterization of $α$-catenaries in terms of their curvature and unit normal vector field.
Smooth axially symmetric Helfrich topological spheres are either round or else they must satisfy a second order equation known as the reduced membrane equation [17]. In this paper, we show that, conversely, axially symmetric closed genus zero solutions of the reduced membrane equation which, in addition, satisfy a rescaling condition are axially symmetric Helfrich spheres. We also exploit this characterization to geometrically describe these surfaces and present convincing evidence that they are symmetric with respect to a suitable plane orthogonal to the axis of rotation and that they belong to a particular infinite discrete family of surfaces.
In this paper, we extend the notion of stationary curves with respect to the moment of inertia from a point $N$ in the Euclidean plane $\mathbb{R}^2$ to the case that the ambient space is either the hyperbolic plane $\mathbb{H}^2$ or the sphere $\mathbb{S}^2$. We characterize the critical points of this energy in terms of the curvature of the curve and the distance to $N$. In $\mathbb{H}^2$, we prove that the only closed stationary curves are circles centered at $N$. In $\mathbb{S}^2$, we estimate the value of $α$ for closed curves according to the hemisphere of $\mathbb{S}^2$ in which the curve lies. In addition, we find the first integrals of the ODEs that describe the parametrizations of stationary curves in both ambient spaces. Finally, we consider the energy minimization problem for curves connecting two points collinear with $N$, in particular solving the case of geodesics.