Minimal isometric immersions into S^2 x R and H^2 x R

For a given simply connected Riemannian surface Sigma, we relate the problem of finding minimal isometric immersions of Sigma into S^2 x R or H^2 x R to a system of two partial differential equations on Sigma. We prove that a constant intrinsic curvature minimal surface in S^2 x R or H^2 x R is either totally geodesic or part of an associate surface of a certain limit of catenoids in H^2 x R. We also prove that if a non constant curvature Riemannian surface admits a continuous one-parameter family of minimal isometric immersions into S^2 x R or H^2 x R, then all these immersions are associate

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

Half-space theorems for minimal surfaces in Nil_3 and Sol_3

We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil_3 and the Lie group Sol_3 endowed with their left-invariant Riemannian metrics. If S is a properly immersed minimal surface in Nil_3 that lies on one side of some entire minimal graph G, then S is the image of G by a vertical translation. If S is a properly immersed minimal surface in Sol_3 that lies on one side of a special plane, then S is another special plane.Comment: 19 pages, 3 figure

Large effects of boundaries on spin amplification in spin chains

We investigate the effect of boundary conditions on spin amplification in spin chains. We show that the boundaries play a crucial role for the dynamics: A single additional coupling between the first and last spins can macroscopically modify the physical behavior compared to the open chain, even in the limit of infinitely long chains. We show that this effect can be understood in terms of a "bifurcation" in Hilbert space that can give access to different parts of Hilbert space with macroscopically different physical properties of the basis functions, depending on the boundary conditions. On the technical side, we introduce semiclassical methods whose precision increase with increasing chain length and allow us to analytically demonstrate the effects of the boundaries in the thermodynamic limit.Comment: replaced figs. 6,10 and corrected corresponding numerical values for initial slopes, added a new fig.7 and a section on total fidelitie