Exotic Minimal Surfaces

We prove a general fusion theorem for complete orientable minimal surfaces in $\mathbb{R}^3$ with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are produced. More specifically, universal surfaces (i.e., surfaces from which all minimal surfaces can be recovered) and space-filling surfaces with arbitrary genus and no symmetries.Comment: 16 pages, 3 figure

A note on the Gauss map of complete nonorientable minimal surfaces

We construct complete nonorientable minimal surfaces whose Gauss map omits two points of the projective plane. This result proves that Fujimoto's theorem is sharp in nonorientable case.Comment: 8 pages, to appear in Pacific J. Mat

Uniform Approximation by Complete Minimal Surfaces of Finite Total Curvature in R3\mathbb{R}^3

An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in $\mathbb{R}^3$ is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total curvature, that is to say, having finite total curvature on proper regions of finite conformal type. We deal only with the orientable case.Comment: 24 pages, 3 figures, research article. This updated version introduces considerably simplifications of notations and arguments, and includes some improvements of the results. The paper will appear in the Transactions of the American Mathematical Societ

Periodic Maximal surfaces in the Lorentz-Minkowski space \l^3

A maximal surface \sb with isolated singularities in a complete flat Lorentzian 3-manifold $\N$ is said to be entire if it lifts to a (periodic) entire multigraph \tilde{\sb} in \l^3. In addition, \sb is called of finite type if it has finite topology, finitely many singular points and \tilde{\sb} is finitely sheeted. Complete and proper maximal immersions with isolated singularities in $\N$ are entire, and entire embedded maximal surfaces in $\N$ with a finite number of singularities are of finite type. We classify complete flat Lorentzian 3-manifolds carrying entire maximal surfaces of finite type, and deal with the topology, Weierstrass representation and asymptotic behavior of this kind of surfaces. Finally, we construct new examples of periodic entire embedded maximal surfaces in \l^3 with fundamental piece having finitely many singularities.Comment: 27 pages, corrected typos, Lemma 2.5 and Theorem 4.1 change

Minimal surfaces in R3\mathbb{R}^3 properly projecting into R2\mathbb{R}^2

For all open Riemann surface M and real number $\theta \in (0,\pi/4),$ we construct a conformal minimal immersion $X=(X_1,X_2,X_3):M \to \mathbb{R}^3$ such that $X_3+\tan(\theta) |X_1|:M \to \mathbb{R}$ is positive and proper. Furthermore, $X$ can be chosen with arbitrarily prescribed flux map. Moreover, we produce properly immersed hyperbolic minimal surfaces with non empty boundary in $\mathbb{R}^3$ lying above a negative sublinear graph.Comment: 24 pages, 7 figures, to appear in Journal of Differential Geometr

Properness of associated minimal surfaces

We prove that for any open Riemann surface $N$ and finite subset $Z\subset \mathbb{S}^1=\{z\in\mathbb{C}\,|\;|z|=1\},$ there exist an infinite closed set $Z_N \subset \mathbb{S}^1$ containing $Z$ and a null holomorphic curve $F=(F_j)_{j=1,2,3}:N\to\mathbb{C}^3$ such that the map $Y:Z_N\times N\to \mathbb{R}^2,$ $Y(v,P)=Re(v(F_1,F_2)(P)),$ is proper. In particular, $Re(vF):N \to\mathbb{R}^3$ is a proper conformal minimal immersion properly projecting into $\mathbb{R}^2=\mathbb{R}^2\times\{0\}\subset\mathbb{R}^3,$ for all $v \in Z_N.$Comment: 17 pages, 5 figure

Null Curves in C3\mathbb{C}^3 and Calabi-Yau Conjectures

For any open orientable surface $M$ and convex domain $\Omega\subset \mathbb{C}^3,$ there exists a Riemann surface $N$ homeomorphic to $M$ and a complete proper null curve $F:N\to\Omega.$ This result follows from a general existence theorem with many applications. Among them, the followings: For any convex domain $\Omega$ in $\mathbb{C}^2$ there exist a Riemann surface $N$ homeomorphic to $M$ and a complete proper holomorphic immersion $F:N\to\Omega.$ Furthermore, if $D \subset \mathbb{R}^2$ is a convex domain and $\Omega$ is the solid right cylinder $\{x \in \mathbb{C}^2 | {Re}(x) \in D\},$ then $F$ can be chosen so that ${\rm Re}(F):N\to D$ is proper. There exists a Riemann surface $N$ homeomorphic to $M$ and a complete bounded holomorphic null immersion $F:N \to {\rm SL}(2,\mathbb{C}).$ There exists a complete bounded CMC-1 immersion $X:M \to \mathbb{H}^3.$ For any convex domain $\Omega \subset \mathbb{R}^3$ there exists a complete proper minimal immersion $(X_j)_{j=1,2,3}:M \to \Omega$ with vanishing flux. Furthermore, if $D \subset \mathbb{R}^2$ is a convex domain and $\Omega=\{(x_j)_{j=1,2,3} \in \mathbb{R}^3 | (x_1,x_2) \in D\},$ then $X$ can be chosen so that $(X_1,X_2):M\to D$ is proper. Any of the above surfaces can be chosen with hyperbolic conformal structure.Comment: 20 pages, 4 figures. To appear in Mathematische Annale

Relative parabolicity of zero mean curvature surfaces in R3R^3 and R13R_1^3

If the Lorentzian norm on a maximal surface in the 3-dimensional Lorentz-Minkowski space $R_1^3$ is positive and proper, then the surface is relative parabolic. As a consequence, entire maximal graphs with a closed set of isolated singularities are relative parabolic. Furthermore, maximal and minimal graphs over closed starlike domains in $R_1^3$ and $R^3,$ respectively, are relative parabolic

On harmonic quasiconformal immersions of surfaces in R3\mathbb{R}^3

This paper is devoted to the study of the global properties of harmonically immersed Riemann surfaces in $\mathbb{R}^3.$ We focus on the geometry of complete harmonic immersions with quasiconformal Gauss map, and in particular, of those with finite total curvature. We pay special attention to the construction of new examples with significant geometry.Comment: 27 pages, 7 figures. Minor changues. To appear in Trans. Amer. Math. So

Approximation theory for non-orientable minimal surfaces and applications

We prove a version of the classical Runge and Mergelyan uniform approximation theorems for non-orientable minimal surfaces in Euclidean 3-space R3. Then, we obtain some geometric applications. Among them, we emphasize the following ones: 1. A Gunning-Narasimhan type theorem for non-orientable conformal surfaces. 2. An existence theorem for non-orientable minimal surfaces in R3, with arbitrary conformal structure, properly projecting into a plane. 3. An existence result for non-orientable minimal surfaces in R3 with arbitrary conformal structure and Gauss map omitting one projective direction.Comment: 34 pages, 4 figure