On forbidden sets

In recent literature there are an increasing number of papers where the forbidden sets of difference equations are computed. We review and complete different attempts to describe the forbidden set and propose new perspectives for further research and a list of open problems in this field.Comment: 30 pages, 5 figure

Higgs effective potential in a perturbed Robertson-Walker background

We calculate the one-loop effective potential of a scalar field in a Robertson-Walker background with scalar metric perturbations. A complete set of orthonormal solutions of the perturbed equations is obtained by using the adiabatic approximation for comoving observers. After analyzing the problem of renormalization in inhomogeneous backgrounds, we get the explicit contribution of metric perturbations to the effective potential. We apply these results to the Standard Model Higgs field and evaluate the effects of metric perturbations on the Higgs mass and on its vacuum expectation value. Space-time variations are found, which are proportional to the gravitational slip parameter, with a typical amplitude of the order of $\Delta\phi/\phi\simeq 10^{-11}$ on cosmological scales. We also discuss possible astrophysical signatures in the Solar System and in the Milky Way that could open new possibilities to explore the symmetry breaking sector of the electroweak interactions.Comment: 10 pages. Erratum section included. Published in Phys. Rev.

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

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

The Mass of Virialized Dark Matter Haloes

Virial mass is used as an estimator for the mass of a dark matter halo. However, the commonly used constant overdensity criterion does not reflect the dynamical structure of haloes. Here we analyze dark matter cosmological simulations in order to obtain properties of haloes of different masses focusing on the size of the region with zero mean radial velocity. Dark matter inside this region is stationary, and thus the mass of this region is a much better approximation for the virial mass.Comment: 2 pages, 1 figure, to be published in the ASPCS Proceedings of the 1st Subaru International Conference "Panoramic Views of Galaxy Formation and Evolution" held in Hayama, Japan 11-16 December 200

The correlation of UHECRs with nearby galaxies in the Local Volume

We explore the possibility of a local origin for ultra high energy cosmic rays (UHECRs). Using the catalogue of Karachentsev et al. including nearby galaxies with distances less than 10Mpc (Local Volume), we search for a correlation with the sample of UHECR events released so far by the Pierre Auger collaboration. The counterpart sample selection is performed with variable distance and luminosity cuts which extract the most likely sources in the catalogue. The probability of chance correlation after penalizing for scans is 0.96%, which corresponds to a correlation signal of 2.6\sigma. We find that the parameters that maximize the signal are \psi=3.0deg, D_{max}=4Mpc and M_B=-15 for the maximum angular separation between cosmic rays and galaxy sources, maximum distance to the source, and sources brighter than B-band absolute magnitude respectively. This implies a preference for the UHECRs arrival directions to be correlated with the nearest and most luminous galaxies in the Local Volume. We note that nearby galaxies with D<10Mpc show a similar correlation with UHECRs as compared to that found by The Pierre Auger Collaboration using active galactic nuclei (AGNs) within 70-100Mpc instead of local galaxies, although less than 20% of cosmic ray events are correlated to a source in our study. However, the observational evidence for mixed composition in the high-energy end of the cosmic ray spectrum supports the possibility of a local origin for UHECRs, as CNO nuclei can travel only few Mpc without strong attenuation by the GZK effect, whereas the observed suppression in the energy spectrum would require more distant sources in the case of pure proton composition interacting with the CMB.Comment: 6 pages, 2 figures, submitted for publication in MNRA

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

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