Complete Constant Mean Curvature surfaces and Bernstein type Theorems in M2×R\mathbb{M}^2\times \mathbb{R}

In this paper we study constant mean curvature surfaces $\Sigma$ in a product space, $\mathbb{M}^2\times \mathbb{R}$, where $\mathbb{M}^2$ is a complete Riemannian manifold. We assume the angle function \nu = \meta{N}{\partial_t} does not change sign on $\Sigma$. We classify these surfaces according to the infimum $c(\Sigma)$ of the Gaussian curvature of the projection of $\Sigma$. When $H \neq 0$ and $c(\Sigma)\geq 0$, then $\Sigma$ is a cylinder over a complete curve with curvature 2H. If H=0 and $c(\Sigma) \geq 0$, then $\Sigma$ must be a vertical plane or $\Sigma$ is a slice $\mathbb{M}^2 \times {t}$, or $\mathbb{M}^2 \equiv \mathbb{R}^2$ with the flat metric and $\Sigma$ is a tilted plane (after possibly passing to a covering space). When $c(\Sigma)\sqrt{-c(\Sigma)} /2$, then $\Sigma$ is a vertical cylinder over a complete curve of $\mathbb{M}^2$ of constant geodesic curvature $2H$. This result is optimal. We also prove a non-existence result concerning complete multi-graphs in $\mathbb{M}^2\times \mathbb{R}$, when $c(\mathbb{M}^2)<0$

Global well-posedness and scattering for the defocusing energy-critical nonlinear Schr\"odinger equation in R1+4\R^{1+4}

We obtain global well-posedness, scattering, uniform regularity, and global $L^6_{t,x}$ spacetime bounds for energy-space solutions to the defocusing energy-critical nonlinear Schr\"odinger equation in $\R\times\R^4$. Our arguments closely follow those of Colliander-Keel-Staffilani-Takaoka-Tao, though our derivation of the frequency-localized interaction Morawetz estimate is somewhat simpler. As a consequence, our method yields a better bound on the $L^6_{t,x}$-norm

Chronology for climate change: Developing age models for the biogeochemical ocean flux study cores

We construct age models for a suite of cores from the northeast Atlantic Ocean by means of accelerator mass spectrometer dating of a key core, BOFS 5K, and correlation with the rest of the suite. The effects of bioturbation and foraminiferal species abundance gradients upon the age record are modeled using a simple equation. The degree of bioturbation is estimated by comparing modeled profiles with dispersal of the Vedde Ash layer in core 5K, and we find a mixing depth of roughly 8 cm for sand-sized material. Using this value, we estimate that age offsets between unbioturbated sediment and some foraminifera species after mixing may be up to 2500 years, with lesser effect on fine carbonate (<10 mu m) ages. The bioturbation model illustrates problems associated with the dating of ''instantaneous'' events such as ash layers and the ''Heinrich'' peaks of ice-rafted detritus. Correlations between core 5K and the other cores from the BOFS suite are made on the basis of similarities in the downcore profiles of oxygen and carbon isotopes, magnetic susceptibility, water and carbonate content, and via marker horizons in X radiographs and ash beds

Discovery of GRS 1915+105 variability patterns in the Rapid Burster

We report the discovery of two new types of variability in the neutron star low-mass X-ray binary MXB 1730-335 (the 'Rapid Burster'). In one observation in 1999, it exhibits a large-amplitude quasi-periodic oscillation with a period of about 7 min. In another observation in 2008, it exhibits two 4-min long 75 per cent deep dips 44 min apart. These two kinds of variability are very similar to the so-called $\rho$ or 'heartbeat' variability and the $\theta$ variability, respectively, seen in the black hole low-mass X-ray binaries GRS 1915+105 and IGR J17091-3624. This shows that these types of behavior are unrelated to a black hole nature of the accretor. Our findings also show that these kinds of behaviour need not take place at near-Eddington accretion rates. We speculate that they may rather be related to the presence of a relatively wide orbit with an orbital period in excess of a few days and about the relation between these instabilities and the type II bursts.Comment: Accepted for publication in MNRAS letter