Superficies mínimas: estabilidad y grupos continuos de isometrías

Resum

A surface in R3 is minimal if and only if it is a critical point of the area functional for any variation with compact support. One of the most interesting properties of a minimal surface is stability. When the surface is minimal and the second derivative of the area functional is non-negative for any variation with compact support, we will say that our minimal surface is stable. Complete stable minimal immersions in R^3 are planes. But if we allow our surfaces to have branch points, then more examples appear. In the first result of this thesis, we study non-orientable, finitely branched, complete, minimal surfaces X : R3 with = C \ E, where E is finite, that are stable. We exhibit a family of such surfaces, {H_n}, called generalized Henneberg surfaces. Theorem A. For every m € N there exists a non-orientable, complete, stable, minimal surface H_m R^3 which has m + 1 branch points. In order to describe other examples, to keep computations simple, we restrict ourselves to the family of minimal surfaces with a single end, and prove that a general Weierstrass data gives rise to a non-orientable, finitely branched, complete, stable minimal surface if and only if some compatibility equations are verified. In terms of the number of branch points m+1, we find: Theorem B. The Henneberg surface H_1 is the only surface with m = 1 that solves the period problem and descends to a 1-sided quotient. For higher complexity, the above uniqueness result fails to hold. We present a 1-parameter family { ; [0, 1)}. This family contains H2 when 0 = 0 and the limit when 0 - 1, after rescaling, is the classical Henneberg surface. In Chapter 3 we describe a new family of free boundary minimal hypersurfaces in the Schwarzschild space. We study the existence of a new family of free boundary minimal hypersurfaces with rotational symmetry in the Schwarzschild space when the dimension 2 around N is large enough. Theorem C. Given m > 0, there exists n_0 N such that for all n > n_0 there exists (n,m) and a family of hypersurfaces { _{t_0}}, t_0 [0, (n,m)) verifying that for t = 0 we have the hypersurface _0 = {p R^n; |p| R_0, x_n(p) = 0} and for every t_0 (0, (n,m)), _{t_0} is a properly embedded, rotationally symmetric, free boundary minimal hypersurface in the Schwarzschild space of dimension n. The techniques we will use to prove Theorem C are based on analyzing the ODE that expresses minimality of a rotationally symmetric hypersurface in the Schwarzschild space and producing a complete solution of this ODE that generates by rotation the desired hypersurface. In the k Schwarzschild spaces, we prove a generalization of the result by R. Montezuma in which he cumputes the Morse index of a plane passing through the origin _0, assuming some extra conditions on the parameters m, k that define a k Schwarzschild spaces. Theorem D. If k (1, 6/5], then the Morse index of _0 in the k Schwarzschild space as a free boundary minimal surface is one for all m > 0. Moreover, if we choose m sufficiently large, then for all k > 1, the Morse index of _0 is one. So far, the ambient riemannian manifolds have infinitely many isometries, which makes the study of their minimal hypersurfaces more tractable than the general case. Another situation in which we have abundance of ambient isometries is that of homogeneous manifolds. A Riemannian manifold (M, g) is homogeneous if for any two points p, q M there is an isometry f Iso(M, g) such that f(p) = q. Among homogeneous manifolds, a remarkable subfamily are those given by Lie groups endowed with left invariant metrics, called metric Lie groups. In low dimension, and assuming that the manifold is simply-connected, these examples of algebraic nature are almost the only possible ones: In dimension 2, the simply-connected homogeneous manifolds reduce to the plane R^2, the hyperbolic plane H^2 and the sphere S^2, and this last one is the only one not being a Lie group. In dimension 3, except for the product manifold S^2 ×R, every simply connected homogeneous 3-manifold is isometric to a metric Lie group. Among three-dimensional, simply-connected metric Lie groups, the semidirect products of the form R^2 _A R for a matrix A M_2(R), are specially relevant since they cover all cases except for the special unitary group SU(2) and the universal cover of the special linear group SL(2,R). In the last chapter, we focus on unimodular metric Lie groups which can be written as a semidirect product, which means that trace(A) = 0. The special role of the vertical direction in R^2 _A R justifies the study of minimal surfaces in R2 _A R which are invariant by left translations by elements in the vertical axis {(0, 0, z); z R}. Every such surface _ R^2 _A R is described by a generating curve of the form { (t) = (x(t), y(t), 0); t I} R2 _A {0}, where I is an open interval in R and is a regular curve. Without lost of generality, we can assume that is parameterized by its arc length (the induced metric on R^2 _A {0} by the natural one on R^2 _A R is the standard euclidean metric on {z = 0}), and thus there exists a function (t) verifying x (t) = cos (t), y (t) = sin (t), t I. Imposing that minimality reduces to a system of differential equations for x(t), y(t), (t). There are previous works that use this approach to describe minimal surfaces which are invariant under a one-parameter group of isometries in Sol_3, the orientation-preserving group of rigid motions in the Lorentz-Minkowski plane. It turns out that Sol_3 can be written as a semidirect product. Other metric Lie groups that can also be written as semidirect products are the Heisenberg space Nil_3, and the universal cover of the group of orientation-preserving rigid motions of the Euclidean plane E(2). From another point of view, not using semidirect products, the results we obtain for Nil_3, where obtained by Figueroa, Mercuri and Pedrosa. The case of the metric Lie group e E(2) carries a one parameter family of metrics of which a particular case is the standard flat one. Even in this standard case, the study of vertically invariant minimal surfaces in E(2) is relevant because of the algebraic behaviour of the minimal surfaces we are studying. When we fix the flat metric on e E(2), we get a technical result that describes this family of minimal surfaces. Some of the properties we give are also preserved when we consider E(2) with a non-flat left invariant metric. Going back to E(2) with its flat metric, we also describe the vertically invariant surfaces with zero Gaussian curvature.