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

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

Complete bounded embedded complex curves in C^2

We prove that any convex domain of C^2 carries properly embedded complete complex curves. In particular, we exhibit the first examples of complete bounded embedded complex curves in C^2Comment: To appear in J. Eur. Math. Soc. (JEMS

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

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

The phi(1020) a0(980) S-wave scattering and hints for a new vector-isovector resonance

We have studied the phi(1020)a0(980) S-wave scattering at threshold energies employing chiral Lagrangians coupled to vector mesons by minimal coupling. The interaction is described without new free parameters by considering the scalar isovector a0(980) resonance as dynamically generated in coupled channels, and demanding that the recently measured e+ e- -> phi(1020) f0(980) cross section is reproduced. For some realistic choices of the parameters, the presence of a dynamically generated isovector companion of the Y(2175) is revealed. We have also investigated the corrections to the e+ e- -> phi(1020) pi0 eta reaction cross section that arise from phi(1020)a0(980) re-scattering in the final state. They are typically large and modify substantially the cross section. For a suitable choice of parameters, the presence of the resonance would manifest itself as a clear peak at sqrt{s}~2.03 GeV in e+ e- -> phi(1020) pi0 eta.Comment: 16 pages, 9 figures, 2 table

The strangeness content of the nucleon from effective field theory and phenomenology

We revisit the classical relation between the strangeness content of the nucleon, the pion-nucleon sigma term and the $SU(3)_F$ breaking of the baryon masses in the context of Lorentz covariant chiral perturbation theory with explicit decuplet-baryon resonance fields. We find that a value of the pion-nucleon sigma term of $\sim$60 MeV is not necessarily at odds with a small strangeness content of the nucleon, in line with the fulfillment of the OZI rule. Moreover, this value is indeed favored by our next-to-leading order calculation. We compare our results with earlier ones and discuss the convergence of the chiral series as well as the uncertainties of chiral approaches to the determination of the sigma terms.Comment: V2 accepted for publication in Physics Letters

Every meromorphic function is the Gauss map of a conformal minimal surface

Let $M$ be an open Riemann surface. We prove that every meromorphic function on $M$ is the complex Gauss map of a conformal minimal immersion $M\to\mathbb{R}^3$ which may furthermore be chosen as the real part of a holomorphic null curve $M\to\mathbb{C}^3$. Analogous results are proved for conformal minimal immersions $M\to\mathbb{R}^n$ for any $n>3$. We also show that every conformal minimal immersion $M\to\mathbb{R}^n$ is isotopic through conformal minimal immersions $M\to\mathbb{R}^n$ to a flat one, and we identify the path connected components of the space of all conformal minimal immersions $M\to\mathbb{R}^n$ for any $n\ge 3$.Comment: J. Geom. Anal., to appear. Available on SpringerLink: https://link.springer.com/article/10.1007%2Fs12220-017-9948-

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.Comment: 10 page