The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space \l^3

We show that a complete embedded maximal surface in the 3-dimensional Lorentz-Minkowski space $L^3$ with a finite number of singularities is, up to a Lorentzian isometry, an entire graph over any spacelike plane asymptotic to a vertical half catenoid or a horizontal plane and with conelike singular points. We study the space $G_n$ of entire maximal graphs over $\{x_3=0\}$ in $L^3$ with $n+1 \geq 2$ conelike singularities and vertical limit normal vector at infinity. We show that $G_n$ is a real analytic manifold of dimension $3n+4,$ and the coordinates are given by the position of the singular points in $R^3$ and the logarithmic growth at the end. We also introduce the moduli space $M_n$ of {\em marked} graphs with $n+1$ singular points (a mark in a graph is an ordering of its singularities), which is a $(n+1)$-sheeted covering of $G_n.$ We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient space $M_n$ is an analytic manifold of dimension $3n-1.$Comment: 32 pages, 4 figures, corrected typos, former Theorem 3.3 (now Theorem 2.2) modifie

The abundance of 28Si32S, 29Si32S, 28Si34S, and 30Si32S in the inner layers of the envelope of IRC+10216

We present high spectral resolution mid-IR observations of SiS towards the C-rich AGB star IRC+10216 carried out with the Texas Echelon-cross-Echelle Spectrograph mounted on the NASA Infrared Telescope Facility. We have identified 204 ro-vibrational lines of 28Si32S, 26 of 29Si32S, 20 of 28Si34S, and 15 of 30Si32S in the frequency range 720-790 cm-1. These lines belong to bands v=1-0, 2-1, 3-2, 4-3, and 5-4, and involve rotational levels with Jlow<90. About 30 per cent of these lines are unblended or weakly blended and can be partially or entirely fitted with a code developed to model the mid-IR emission of a spherically symmetric circumstellar envelope composed of expanding gas and dust. The observed lines trace the envelope at distances to the star <35R* (~0.7 arcsec). The fits are compatible with an expansion velocity of 1+2.5(r/R*-1) km/s between 1 and 5R*, 11 km/s between 5 and 20R*, and 14.5 km/s outwards. The derived abundance profile of 28Si32S with respect to H2 is 4.9e-6 between the stellar photosphere and 5R*, decreasing linearly to 1.6e-6 at 20R* and to 1.3e-6 at 50R*. 28Si32S seems to be rotationally under LTE in the region of the envelope probed with our observations and vibrationally out of LTE in most of it. There is a red-shifted emission excess in the 28Si32S lines of band v=1-0 that cannot be found in the lines of bands v=2-1, 3-2, 4-3, and 5-4. This excess could be explained by an enhancement of the vibrational temperature around 20R* behind the star. The derived isotopic ratios 28Si/29Si, and 32S/34S are 17 and 14, compatible with previous estimates.Comment: 11 pages, 5 figures, and 4 tables. Accepted for publication in MNRA