Superficies mínimas y de curvatura media constante en espacios homogéneos

Abstract

The theory of minimal and constant mean curvature surfaces in the Euclidean space R3 is a classical field in Differential Geometry. It gathers different techniques such as Complex Analysis, Geometry Measure Theory and Partial Differential Equations, as well as Topology and Algebra. Nowadays it is still an important research field with different applications in Geometry and other areas of Mathematics. The origins of minimal surfaces date back to 1760 when Lagrange proposed the problem, previously treated by Euler for revolution surfaces, of finding a surface with the least area possible which encloses a given closed curve without self-intersections. This approach was subsequently expanded by the experimental model of the physicist Plateau, that consists in immersing a closed curved of thin wire in soapy water. Removing the wire carefully, the solution to this problem appears, which has in general, the shape of a regular surface and remain still by the action of the surface tension of the liquid. By the Laplace-Young Law, such a surface has mean curvature zero. Surfaces with constant mean curvature equal to zero are known as minimal surfaces. Lagrange’s problem is an example of the method that it is known as Calculus of Variations nowadays. Area-minimizing surfaces, that is, those that solve the minimization problem, are critical points of the Area functional (which is equivalent to being a minimal surface), though not every critical point of this functional solves the minimization problem. We can argue that the minimal surfaces which are observable are locally minima of the Area functional. In particular, they are stable, that is, the second derivate of the Area functional is bigger than o equal zero for all compactly supported variation of the surface.