Plateau's problem in Finsler 3-space

We explore a connection between the Finslerian area functional based on the Busemann-Hausdorff-volume form, and well-investigated Cartan functionals to solve Plateau's problem in Finsler 3-space, and prove higher regularity of solutions. Free and semi-free geometric boundary value problems, as well as the Douglas problem in Finsler space can be dealt with in the same way. We also provide a simple isoperimetric inequality for minimal surfaces in Finsler spaces.Comment: 42 page

On minimal immersions in Finsler spaces

We explore a connection between the Finslerian area functional and well-investigated Cartan functionals to prove new Bernstein theorems, uniqueness and removability results for Finsler-minimal graphs, as well as enclosure theorems and isoperimetric inequalities for minimal immersions in Finsler spaces. In addition, we establish the existence of smooth Finsler-minimal immersions spanning given extreme or graphlike boundary contours.Comment: 26 pages, changed numbering of equation

On sphere-filling ropes

What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope's thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seamlines of a tennis ball, others exhibit a striking resemblance to Turing patterns in chemistry, or to ordered phases of long elastic rods stuffed into spherical shells.Comment: 15 pages, 8 figure

On some knot energies involving Menger curvature

We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. All of these energies turn out to be charge, minimizable in given isotopy classes, tight and strong. Almost all distinguish between knots and unknots, and some of them can be shown to be uniquely minimized by round circles. Bounds on the stick number and the average crossing number, some non-trivial global lower bounds, and unique minimization by circles upon compaction complete the picture.Comment: 31 pages, 4 figures; version 2 with minor changes and modification

Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies

In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded $m$-dimensional Lipschitz submanifolds in ${\mathbb{R}}^n$. It turns out that due to a smoothing effect any sequence of submanifolds with uniformly bounded energy contains a subsequence converging in $C^1$ to a limit submanifold. This result has two applications. The first one is an isotopy finiteness theorem: there are only finitely many isotopy types of such submanifolds below a given energy value, and we provide explicit bounds on the number of isotopy types in terms of the respective energy. The second one is the lower semicontinuity - with respect to Hausdorff-convergence of submanifolds - of all geometric curvature energies under consideration, which can be used to minimise each of these energies within prescribed isotopy classes.Comment: 44 pages, 5 figure

What are the Longest Ropes on the Unit Sphere?

We consider the variational problem of finding the longest closed curves of given minimal thickness on the unit sphere. After establishing the existence of solutions for any given thickness between 0 and 1, we explicitly construct for each given thickness $${\Theta_n:= {\rm sin}\, \pi/(2n),}$$ $${n\in\mathbb{N}}$$, exactly $${\varphi(n)}$$ solutions, where $${\varphi}$$ is Euler's totient function from number theory. Then we prove that these solutions are unique, and also provide a complete characterisation of sphere filling curves on the unit sphere; that is of those curves whose spherical tubular neighbourhood completely covers the surface area of the unit sphere exactly once. All of these results carry over to open curves as well, as indicated in the last sectio

Tiempos del golpismo latinoamericano

Publicado en: Historia y Política, 5: 7-27, 2001[EN] Military interventions in Latin American politics had been often requested by political and social actors, unsatisfied with electoral or governmental results. After the Cuban revolution the foreign dimension acquires a new relevance, as ideological polarization inside many countries increases the US influence on Latin American armies. After a new change in Washington’s Latin American policy, during the Carter administration, and the devastating effects of civil wars and violations of human rights, in the last years the social and political climate has been clearly opposite to new military interventions.[ES] Las intervenciones militares en América Latina han sido a menudo una consecuencia de la insatisfacción de los actores sociales y políticos ante los gobiernos o los resultados electorales. La dimensión exterior, y en concreto la influencia norteamericana, comienza a ser decisiva durante el período de polarización ideológica que sigue a la revolución cubana. El nuevo giro de la política exterior norteamericana, durante la administración Carter, y las terribles consecuencias de las guerras civiles y violaciones de los derechos humanos, han creado durante los últimos años un clima político y social claramente opuesto a nuevas intervenciones militares.Peer reviewe