The number of conformally equivalent maximal graphs

We show that the number of entire maximal graphs with finitely many singular points that are conformally equivalent is a universal constant that depends only on the number of singularities, namely 2n for graphs with n + 1 singularities. We also give an explicit description of the family of entire maximal graphs with a finite number of singularities all of them lying on a plane orthogonal to the limit normal vector at infinity.Ministerio de Educación y Ciencia MTM2007-64504Junta de Andalucía P06-FQM-01642Junta de Andalucía FQM32

Cotunneling theory of inelastic STM spin spectroscopy

We propose cotunneling as the microscopic mechanism that makes possible inelastic electron spectroscopy of magnetic atoms in surfaces for a wide range of systems, including single magnetic adatoms, molecules and molecular stacks. We describe electronic transport between the scanning tip and the conducting surface through the magnetic system (MS) with a generalized Anderson model, without making use of effective spin models. Transport and spin dynamics are described with an effective cotunneling Hamiltonian in which the correlations in the magnetic system are calculated exactly and the coupling to the electrodes is included up to second order in the tip-MS and MS-substrate. In the adequate limit our approach is equivalent to the phenomenological Kondo exchange model that successfully describe the experiments . We apply our method to study in detail inelastic transport in two systems, stacks of Cobalt Phthalocyanines and a single Mn atom on Cu$_2$N. Our method accounts both, for the large contribution of the inelastic spin exchange events to the conductance and the observed conductance asymmetry.Comment: 12 pages, 6 figure

Constant mean curvature surfaces in 3-dimensional Thurston geometries

This is a survey on the global theory of constant mean curvature surfaces in Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight canonical Thurston 3-dimensional geometries, i.e. R3, H3, S3, H2 × R, S2 × R, the Heisenberg space Nil3, the universal cover of PSL2(R) and the Lie group Sol3. We will focus on the problems of classifying compact CMC surfaces and entire CMC graphs in these spaces. A collection of important open problems of the theory is also presented.Ministerio de Educación y Ciencia MTM2007-65249Junta de Andalucía FQM325Junta de Andalucía P06-FQM-0164

Complete minimal surfaces in R3 with a prescribed coordinate function

In this paper we construct complete simply connected minimal surfaces with a prescribed coordinate function. Moreover, we prove that these surfaces are dense in the space of all minimal surfaces with this coordinate function (with the topol- ogy of the smooth convergence on compact sets).Ministerio de Educación y Ciencia MTM2007-61775Ministerio de Educación y Ciencia MTM2007-6450

On the uniqueness of the helicoid and Enneper’s surface in the Lorentz-Minkowski space R31

In this paper we deal with the uniqueness of the Lorentzian helicoid and Enneper’s surface among properly embedded maximal surfaces with lightlike boundary of mirror symmetry in the Lorentz-Minkowski space R3Ministerio de Ciencia y Tecnología MTM2004-00160Ministerio de Ciencia y Tecnología MTM2007-61775Junta de Andalucía P06-FQM-01642Junta de Andalucía FQM32

Totally umbilical disks and applications to surfaces in three-dimensional homogeneous spaces

Following [Ch] and [dCF], we give sufficient conditions for a disk type surface, with piecewise smooth boundary, to be totally umbilical for a given Coddazi pair. As a consequence, we obtain rigidity results for surfaces in space forms and in homogeneous product spaces that generalizes some known results.Ministerio de Educación y Ciencia MTM2007-65249Ministerio de Educación y Ciencia MTM2007-64504Junta de Andalucía P06-FQM-01642Junta de Andalucía FQM32

Derivation of the spin Hamiltonians for Fe in MgO

A method to calculate the effective spin Hamiltonian for a transition metal impurity in a non- magnetic insulating host is presented and applied to the paradigmatic case of Fe in MgO. In a first step we calculate the electronic structure employing standard density functional theory (DFT), based on generalized-gradient approximation (GGA), using plane waves as a basis set. The corresponding basis of atomic-like maximally localized Wannier functions is derived and used to represent the DFT Hamiltonian, resulting in a tight-binding model for the atomic orbitals of the magnetic impurity. The third step is to solve, by exact numerical diagonalization, the N electron problem in the open shell of the magnetic atom, including both effect of spin-orbit and Coulomb repulsion. Finally, the low energy sector of this multi-electron Hamiltonian is mapped into effective spin models that, in addition to the spin matrices S, can also include the orbital angular momentum L when appropriate. We successfully apply the method to Fe in MgO, considering both, the undistorted and Jahn-Teller (JT) distorted cases. Implications for the influence of Fe impurities on the performance of magnetic tunnel junctions based on MgO are discussed.Comment: 10 pages, 7 Figure

The Gauss map of surfaces in PSL˜2(R)

We define a Gauss map for surfaces in the universal cover of the Lie group PSL2(R) endowed with a left-invariant Riemannian metric having a 4-dimensional isometry group. This Gauss map is not related to the Lie group structure. We prove that the Gauss map of a nowhere vertical surface of critical constant mean curvature is harmonic into the hyperbolic plane H2 and we obtain a Weierstrass-type representation formula. This extends results in H2 ×R and the Heisenberg group Nil3, and completes the proof of existence of harmonic Gauss maps for surfaces of critical constant mean curvature in any homogeneous manifold diffeomorphic to R3 with isometry group of dimension at least 4.Ministerio de Ciencia y Tecnología MTM2010-19821Junta de Andalucía P09-FQM-508

Harmonic mappings and conformal minimal immersions of Riemann surfaces into RN

We prove that for any open Riemann surface N, natural number N ≥ 3, non-constant harmonic map h:N→R N−2 and holomorphic 2-form H on N , there exists a weakly complete harmonic map X=(Xj)j=1,…,\scN:N→R\scN with Hopf differential H and (Xj)j=3,…,\scN=h. In particular, there exists a complete conformal minimal immersion Y=(Yj)j=1,…,\scN:N→R\scN such that (Yj)j=3,…,\scN=h . As some consequences of these results (1) there exist 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 CP\scN−1 in general position. (2) There exist complete non-proper embedded minimal surfaces in R\scN, ∀\scN>3.Ministerio de Ciencia y Tecnología MTM2007-61775Ministerio de Ciencia y Tecnología MTM2007-64504Junta de Andalucía P09-FQM-508