Helicoidal minimal surfaces in the 3-sphere: An approach via spherical curves

We prove an existence and uniqueness theorem about spherical helicoidal (in particular, rotational) surfaces with prescribed mean or Gaussian curvature in terms of a continuous function depending on the distance to its axis. As an application in the case of vanishing mean curvature, it is shown that the well-known conjugation between the belicoid and the catenoid in Euclidean three-space extends naturally to the 3-sphere to their spherical versions and determine in a quite explicit way their associated surfaces in the sense of Lawson. As a key tool, we use the notion of spherical angular momentum of the spherical curves that play the role of profile curves of the minimal helicoidal surfaces in the 3-sphere.Comment: 22 pages, 4 figure

Curves in the Lorentz-Minkowski plane with curvature depending on their position

Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in the Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in 2 whose curvature depends on the Lorentzian pseudodistance from the origin, and those ones whose curvature depends on the Lorentzian pseudodistance through the horizontal or vertical geodesic to a fixed lightlike geodesic. Making use of the notions of geometric angular momentum (with respect to the origin) and geometric linear momentum (with respect to the fixed lightlike geodesic), respectively, we get two abstract integrability results to determine such curves through quadratures. In this way, we find out several new families of Lorentzian spiral, special elastic and grim-reaper curves whose intrinsic equations are expressed in terms of elementary functions. In addition, we provide uniqueness results for the generatrix curve of the Enneper surface of second kind and for Lorentzian versions of some well-known curves in the Euclidean setting, like the Bernoulli lemniscate, the cardioid, the sinusoidal spirals and some nondegenerate conics. We are able to get arc-length parametrizations of them and they are depicted graphically.European Union (EU) Spanish Government MTM2017-89677-PMECD FPU16/0309

Coloreando triángulos con sentido

Se propone un problema que sirva para motivar y afianzar el estudio de la trigonometría plana y para poder experimentar y sentir las matemáticas asignando “con sentido” un color del modelo RGB a cualquier triángulo del plano. El modo propuesto consiste en elegir una terna de números que provienen de los cocientes (dos a dos) de los lados del triángulo una vez ordenados de menor a mayor; cada término de la terna indicará el peso en rojo (Red), verde (Green) y azul (Blue) que componen el color asignado al triángulo en cuestión. Se discute la bondad de la definición anterior y se ilustra con detalle y por separado en las clases de triángulos rectángulos, isósceles, acutángulos y obtusángulos. Acompañamos ilustraciones en color de numerosos ejemplos

Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers

This article is motivated by a problem posed by David A. Singer in 1999 and by the classical Euler elastic curves. We study spacelike and timelike curves in the Lorentz-Minkowski plane 2 whose curvature is expressed in terms of the Lorentzian pseudodistance to fixed geodesics. In this way, we get a complete description of all the elastic curves in 2 and provide the Lorentzian versions of catenaries and grim-reaper curves. We show several uniqueness results for them in terms of their geometric linear momentum. In addition, we are able to get arc-length parametrizations of all the aforementioned curves and they are depicted graphically

A Geometric Application of Runge's Theorem

En aquest article donem una demostració simple de l'existència d'aplicacions harmòniques de qualsevol superfície de Riemann cap al pla complex C = R2. La nostra eina principal és la teoria d'aproximació per funcions holomorfes en superfícies de Riemann.In this article we give a simple proof of the existence of proper harmonic maps from any open Riemann surface into the complex plane C=R^2. Our main tool will be the Approximation Theory by holomorphic functions on Riemann surfaces

Teoría de Interpolación por Superficies Mínimas

Tesis Univ. Granada. Programade Doctorado en MatemáticasEsta tesis está parcialmente financiado por la Agencia Estatal de Investigación (SRA) y el Fondo Europeo de Desarrollo Regional (ERDF) vía el proyecto MTM2017-89677-P, MINECO, España; así como por la beca de investigación BES-2015-071993 del Plan Estatal de Investigación Científica y Técnica e Innovación; y el Fondo Social Europeo (FSE)