Problema de Plateau para superfícies mínimas com um fim tipo catenóide

Abstract

The study of Minimal surfaces has a long and rich history, dating back to the beginnings of calculus of variations and taking momentum from the experiments of the Belgian physicist J. Plateau in 1847. He showed that by the laws of surface tension, the soap film formed by dipping a closed wire contour into a soap solution, represents a surface which is stable with respect to the area. That is, under minimal distortion, the film always becomes larger. This problem represented a great challenge because of its contrast between the simplicity of being established and the difficulty of the solution: to find a surface G of smaller area having a given Jordan curve. The mathematical solution of this problem in dimension two was given independently in 1930 and 1931 by T. Radó and J. Douglas for a Jordan curve rectifiable in Euclidean space. The minimum surface that they obtained was realized by a harmonic application as defined in the unit disc. In 1940, R. Courant solved this same problem for minimum surfaces of different topological types. In 1948 Morrey proved Douglas and Radó's theorem in a homogeneously regular Riemannian manifold. A natural question is whether every Jordan curve also limits an infinite "ring of minimum area," ie, a smaller area of ​​the conformal type of the unit disc minus one point which extends to infinity. This is Plateau's Outer Problem. In 1989 Friedrich Tomi and Rugang Ye in [2] showed that any rectifiable Jordan curve in R3 limits a minimal immersion of a ring, which extends to infinity and has a flat end, ie, outside some ball surface is the graph of a limited function, defined in some domain by assonting a plane at infinity. In this work we will solve the corresponding problem for surfaces with catenode type end, which means that in infinity the surface is the graph of a growing logarithmic function, and therefore resembles the form of a catenoid medium.