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Francisco J. Lopez

Publications and source records attributed to Francisco J. Lopez.

34 records · Page 2Linked to original sources
arXiv2012

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 $v \in Z_N.$

math.DG↗
arXiv2012

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

For any open orientable surface $M$ and convex domain $Ω\subset \mathbb{C}^3,$ there exists a Riemann surface $N$ homeomorphic to $M$ and a complete proper null curve $F:N\toΩ.$ This result follows from a general existence theorem with many applications. Among them, the followings: For any convex domain $Ω$ in $\mathbb{C}^2$ there exist a Riemann surface $N$ homeomorphic to $M$ and a complete proper holomorphic immersion $F:N\toΩ.$ Furthermore, if $D \subset \mathbb{R}^2$ is a convex domain and $Ω$ 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 $Ω\subset \mathbb{R}^3$ there exists a complete proper minimal immersion $(X_j)_{j=1,2,3}:M \to Ω$ with vanishing flux. Furthermore, if $D \subset \mathbb{R}^2$ is a convex domain and $Ω=\{(x_j)_{j=1,2,3} \in \mathbb{R}^3 | (x_1,x_2) \in D\},$ 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.

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arXiv2012

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

For all open Riemann surface M and real number $θ\in (0,π/4),$ we construct a conformal minimal immersion $X=(X_1,X_2,X_3):M \to \mathbb{R}^3$ such that $X_3+\tan(θ) |X_1|:M \to \mathbb{R}$ 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.

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arXiv2011

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

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.

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arXiv2011

Compact complete null curves in Complex 3-space

We prove that for any open orientable surface $S$ of finite topology, there exist a Riemann surface $\mathcal{M},$ a relatively compact domain $M\subset\mathcal{M}$ and a continuous map $X:\bar{M}\to\mathbb{C}^3$ such that: $\mathcal{M}$ and $M$ are homeomorphic to $S,$ $\mathcal{M}-M$ and $\mathcal{M}-\bar{M}$ contain no relatively compact components in $\mathcal{M},$ $X|_M$ is a complete null holomorphic curve, $X|_{\bar{M}-M}:\bar{M}-M\to\mathbb{C}^3$ is an embedding and the Hausdorff dimension of $X(\bar{M}-M)$ is $1.$ Moreover, for any $ε>0$ and compact null holomorphic curve $Y:N\to\mathbb{C}^3$ with non-empty boundary $Y(\partial N),$ there exist Riemann surfaces $M$ and $\mathcal{M}$ homeomorphic to $N^\circ$ and a map $X:\bar{M}\to\mathbb{C}^3$ in the above conditions such that $δ^H(Y(\partial N),X(\bar{M}-M))<ε,$ where $δ^H(\cdot,\cdot)$ means Hausdorff distance in $\mathbb{C}^3.$

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arXiv2010

Harmonic mappings and conformal minimal immersions of Riemann surfaces into $\mathbb{R}^n$

We prove that for any open Riemann surface $N,$ natural number $n\geq 3,$ non-constant harmonic map $h:N\to \mathbb{R}^{n-2}$ and holomorphic 2-form $H$ on $N,$ there exists a weakly complete harmonic map $X=(X_j)_{j=1,\ldots,n}:N \to \mathbb{R}^n$ with Hopf differential $H$ and $(X_j)_{j=3,\ldots,n}=h.$ In particular, there exists a complete conformal minimal immersion $Y=(Y_j)_{j=1,\ldots,n}:N \to \mathbb{R}^n$ such that $(Y_j)_{j=3,\ldots,n}=h.$ As a consequence of these results, complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect $n$ hyperplanes of $\mathbb{CP}^{n-1}$ in general position are constructed. Moreover, complete non-proper embedded minimal surfaces in $\mathbb{R}^n,$ $\forall n>3,$ are exhibited.

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arXiv2010

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.

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arXiv2010

Nonorientable maximal surfaces in the Lorentz-Minkowski 3-space

The geometry and topology of complete nonorientable maximal surfaces with lightlike singularities in the Lorentz-Minkowski 3-space are studied. Some topological congruence formulae for surfaces of this kind are obtained. As a consequence, some existence and uniqueness results for maximal Moebius strips and maximal Klein bottles with one end are proved.

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arXiv2009

Complete minimal surfaces and harmonic functions

We prove that for any open Riemann surface $M$ and any non constant harmonic function $h:M \to \mathbb{R},$ there exists a complete conformal minimal immersion $X:M \to \mathbb{R}^3$ whose third coordinate function coincides with $h.$ As a consequence, complete minimal surfaces with arbitrary conformal structure and whose Gauss map misses two points are constructed.

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arXiv2005

The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space $ł^3$

We show that a complete embedded maximal surface in the 3-dimensional Lorentz-Minkowski space $L^3$ with a finite number of singularities is, up to a Lorentzian isometry, an entire graph over any spacelike plane asymptotic to a vertical half catenoid or a horizontal plane and with conelike singular points. We study the space $G_n$ of entire maximal graphs over $\{x_3=0\}$ in $L^3$ with $n+1 \geq 2$ conelike singularities and vertical limit normal vector at infinity. We show that $G_n$ is a real analytic manifold of dimension $3n+4,$ and the coordinates are given by the position of the singular points in $R^3$ and the logarithmic growth at the end. We also introduce the moduli space $M_n$ of {\em marked} graphs with $n+1$ singular points (a mark in a graph is an ordering of its singularities), which is a $(n+1)$-sheeted covering of $G_n.$ We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient space $M_n$ is an analytic manifold of dimension $3n-1.$

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arXiv2005

Periodic Maximal surfaces in the Lorentz-Minkowski space $ł^3$

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 $ł^3.$ 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 $ł^3$ with fundamental piece having finitely many singularities.

math.DG↗
arXiv2004

The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz-Minkowski space $ł^3$

We show that, up to some natural normalizations, the moduli space of singly periodic complete embedded maximal surfaces in the Lorentz-Minkowski space $ł^3=(\r^3,dx_1^2+dx_2^2-dx_3^2),$ with fundamental piece having a finite number $(n+1)$ of singularities, is a real analytic manifold of dimension $3n+4.$ The underlying topology agrees with the topology of uniform convergence of graphs on compact subsets of $\{x_3=0\}.$

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arXiv2004

Relative parabolicity of zero mean curvature surfaces in $R^3$ and $R_1^3$

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.

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