Kaehler submanifolds with parallel pluri-mean curvature

We investigate the local geometry of a class of K\"ahler submanifolds $M \subset \R^n$ which generalize surfaces of constant mean curvature. The role of the mean curvature vector is played by the $(1,1)$-part (i.e. the $dz_id\bar z_j$-components) of the second fundamental form $\alpha$, which we call the pluri-mean curvature. We show that these K\"ahler submanifolds are characterized by the existence of an associated family of isometric submanifolds with rotated second fundamental form. Of particular interest is the isotropic case where this associated family is trivial. We also investigate the properties of the corresponding Gauss map which is pluriharmonic.Comment: Plain TeX, 21 page

Parallel Mean Curvature Surfaces in Symmetric Spaces

We present a reduction of codimension theorem for surfaces with parallel mean curvature in symmetric spaces

The Cosmological Time Function

Let $(M,g)$ be a time oriented Lorentzian manifold and $d$ the Lorentzian distance on $M$. The function $\tau(q):=\sup_{p< q} d(p,q)$ is the cosmological time function of $M$, where as usual $p< q$ means that $p$ is in the causal past of $q$. This function is called regular iff $\tau(q) < \infty$ for all $q$ and also $\tau \to 0$ along every past inextendible causal curve. If the cosmological time function $\tau$ of a space time $(M,g)$ is regular it has several pleasant consequences: (1) It forces $(M,g)$ to be globally hyperbolic, (2) every point of $(M,g)$ can be connected to the initial singularity by a rest curve (i.e., a timelike geodesic ray that maximizes the distance to the singularity), (3) the function $\tau$ is a time function in the usual sense, in particular (4) $\tau$ is continuous, in fact locally Lipschitz and the second derivatives of $\tau$ exist almost everywhere.Comment: 19 pages, AEI preprint, latex2e with amsmath and amsth

Rigid Singularity Theorem in Globally Hyperbolic Spacetimes

We show the rigid singularity theorem, that is, a globally hyperbolic spacetime satisfying the strong energy condition and containing past trapped sets, either is timelike geodesically incomplete or splits isometrically as space $\times$ time. This result is related to Yau's Lorentzian splitting conjecture.Comment: 3 pages, uses revtex.sty, to appear in Physical Review

Multi-Terabyte EIDE Disk Arrays running Linux RAID5

High-energy physics experiments are currently recording large amounts of data and in a few years will be recording prodigious quantities of data. New methods must be developed to handle this data and make analysis at universities possible. Grid Computing is one method; however, the data must be cached at the various Grid nodes. We examine some storage techniques that exploit recent developments in commodity hardware. Disk arrays using RAID level 5 (RAID-5) include both parity and striping. The striping improves access speed. The parity protects data in the event of a single disk failure, but not in the case of multiple disk failures. We report on tests of dual-processor Linux Software RAID-5 arrays and Hardware RAID-5 arrays using a 12-disk 3ware controller, in conjunction with 250 and 300 GB disks, for use in offline high-energy physics data analysis. The price of IDE disks is now less than $1/GB. These RAID-5 disk arrays can be scaled to sizes affordable to small institutions and used when fast random access at low cost is important.Comment: Talk from the 2004 Computing in High Energy and Nuclear Physics (CHEP04), Interlaken, Switzerland, 27th September - 1st October 2004, 4 pages, LaTeX, uses CHEP2004.cls. ID 47, Poster Session 2, Track

Convex Functions and Spacetime Geometry

Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime $(M,g_{\mu \nu})$ or an initial data set $(\Sigma, h_{ij}, K_{ij})$ admitting a suitably defined convex function. We show how the existence of a convex function on a spacetime places restrictions on the properties of the spacetime geometry.Comment: 26 pages, latex, 7 figures, improved version. some claims removed, references adde

Metrics with Prescribed Ricci Curvature near the Boundary of a Manifold

Suppose $M$ is a manifold with boundary. Choose a point $o\in\partial M$. We investigate the prescribed Ricci curvature equation \Ric(G)=T in a neighborhood of $o$ under natural boundary conditions. The unknown $G$ here is a Riemannian metric. The letter $T$ in the right-hand side denotes a (0,2)-tensor. Our main theorems address the questions of the existence and the uniqueness of solutions. We explain, among other things, how these theorems may be used to study rotationally symmetric metrics near the boundary of a solid torus $\mathcal T$. The paper concludes with a brief discussion of the Einstein equation on $\mathcal T$.Comment: 13 page

Crystal structure of the dynamin tetramer

The mechanochemical protein dynamin is the prototype of the dynamin superfamily of large GTPases, which shape and remodel membranes in diverse cellular processes. Dynamin forms predominantly tetramers in the cytosol, which oligomerize at the neck of clathrin-coated vesicles to mediate constriction and subsequent scission of the membrane. Previous studies have described the architecture of dynamin dimers, but the molecular determinants for dynamin assembly and its regulation have remained unclear. Here we present the crystal structure of the human dynamin tetramer in the nucleotide-free state. Combining structural data with mutational studies, oligomerization measurements and Markov state models of molecular dynamics simulations, we suggest a mechanism by which oligomerization of dynamin is linked to the release of intramolecular autoinhibitory interactions. We elucidate how mutations that interfere with tetramer formation and autoinhibition can lead to the congenital muscle disorders Charcot-Marie-Tooth neuropathy and centronuclear myopathy, respectively. Notably, the bent shape of the tetramer explains how dynamin assembles into a right-handed helical oligomer of defined diameter, which has direct implications for its function in membrane constriction

ff-minimal surface and manifold with positive mm-Bakry-\'{E}mery Ricci curvature

In this paper, we first prove a compactness theorem for the space of closed embedded $f$-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-\'{E}mery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the $f$-Laplacian on compact manifold with positive $m$-Bakry-\'{E}mery Ricci curvature, and prove that the lower bound is achieved only if the manifold is isometric to the $n$-shpere, or the $n$-dimensional hemisphere. Finally, for compact manifold with positive $m$-Bakry-\'{E}mery Ricci curvature and $f$-mean convex boundary, we prove an upper bound for the distance function to the boundary, and the upper bound is achieved if only if the manifold is isometric to an Euclidean ball.Comment: 15 page