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

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

Redundant Arrays of IDE Drives

The next generation of high-energy physics experiments is expected to gather prodigious amounts of data. New methods must be developed to handle this data and make analysis at universities possible. We examine some techniques that use recent developments in commodity hardware. We test redundant arrays of integrated drive electronics (IDE) disk drives for use in offline high-energy physics data analysis. IDE redundant array of inexpensive disks (RAID) prices now equal the cost per terabyte of million-dollar tape robots! The arrays can be scaled to sizes affordable to institutions without robots and used when fast random access at low cost is important. We also explore three methods of moving data between sites; internet transfers, hot pluggable IDE disks in FireWire cases, and writable digital video disks (DVD-R).Comment: Submitted to IEEE Transactions On Nuclear Science, for the 2001 IEEE Nuclear Science Symposium and Medical Imaging Conference, 8 pages, 1 figure, uses IEEEtran.cls. Revised March 19, 2002 and published August 200

Surgery and the Spectrum of the Dirac Operator

We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension at least 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order. The proof is based on a surgery theorem for the Dirac spectrum which says that if one performs surgery of codimension at least 3 on a closed Riemannian spin manifold, then the Dirac spectrum changes arbitrarily little provided the metric on the manifold after surgery is chosen properly.Comment: 23 pages, 4 figures, to appear in J. Reine Angew. Mat

Invariant four-forms and symmetric pairs

We give criteria for real, complex and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations whose second exterior power is irreducible or has an irreducible summand of co-dimension one, and we give a conceptual computation-free argument for the construction of the exceptional Lie algebras of compact type.Comment: 16 pages [v2: references added, last section expanded

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

Volume Comparison for Hypersurfaces in Lorentzian Manifolds and Singularity Theorems

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof of Hawking's singularity theorem.Comment: 15 pages, LaTe