A generalization of Hawking's black hole topology theorem to higher dimensions

Hawking's theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking's results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology $S^2 \times S^1$. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).Comment: 8 pages, latex2e, references updated, minor corrections, to appear in Communications in Mathematical Physic

The Jang equation reduction of the spacetime positive energy theorem in dimensions less than eight

We extend the Jang equation proof of the positive energy theorem due to R. Schoen and S.-T. Yau from dimension $n=3$ to dimensions $3 \leq n <8$. This requires us to address several technical difficulties that are not present when $n=3$. The regularity and decay assumptions for the initial data sets to which our argument applies are weaker than those of R. Schoen and S.-T. Yau. In recent joint work with L.-H. Huang, D. Lee, and R. Schoen we have given a different proof of the full positive mass theorem in dimensions $3 \leq n < 8$. We pointed out that this theorem can alternatively be derived from our density argument and the positive energy theorem of the present paper.Comment: All comments welcome! Final version to appear in Comm. Math. Phy

Existence, Regularity, and Properties of Generalized Apparent Horizons

We prove a conjecture of Tom Ilmanen's and Hubert Bray's regarding the existence of the outermost generalized apparent horizon in an initial data set and that it is outer area minimizing.Comment: 16 pages, thoroughly revised, no major changes, to appear in Comm. Math. Phy

Foliations and Chern-Heinz inequalities

We extend the Chern-Heinz inequalities about mean curvature and scalar curvature of graphs of $C^{2}$-functions to leaves of transversally oriented codimension one $C^{2}$-foliations of Riemannian manifolds. That extends partially Salavessa's work on mean curvature of graphs and generalize results of Barbosa-Kenmotsu-Oshikiri \cite{barbosa-kenmotsu-Oshikiri} and Barbosa-Gomes-Silveira \cite{barbosa-gomes-silveira} about foliations of 3-dimensional Riemannian manifolds by constant mean curvature surfaces. These Chern-Heinz inequalities for foliations can be applied to prove Haymann-Makai-Osserman inequality (lower bounds of the fundamental tones of bounded open subsets $\Omega \subset \mathbb{R}^{2}$ in terms of its inradius) for embedded tubular neighborhoods of simple curves of $\mathbb{R}^{n}$.Comment: This paper is an improvment of an earlier paper titled On Chern-Heinz Inequalities. 8 Pages, Late

Gluing Initial Data Sets for General Relativity

We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completely local in the sense that the initial data is left unaltered on the complement of arbitrarily small neighborhoods of the points about which the gluing takes place. Using this construction we establish the existence of cosmological, maximal globally hyperbolic, vacuum space-times with no constant mean curvature spacelike Cauchy surfaces.Comment: Final published version - PRL, 4 page

The role of string-like, supramolecular assemblies in reentrant supernematic liquid crystals

Using a combination of isothermal-isobaric Monte Carlo and microcanonical molecular dynamics we investigate the relation between structure and self-diffusion in various phases of a model liquid crystal using the Gay-Berne-Kihara potential. These molecules are confined to a mesoscopic slit-pore with atomically smooth substrate surfaces. As reported recently [see M. G. Mazza {\em et al.}, Phys. Rev. Lett. {\bf 105}, 227802 (2010)], a reentrant nematic (RN) phase may form at sufficiently high pressures/densities. This phase is characterized by a high degree of nematic order and a substantially enhanced self-diffusivity in the direction of the director $\hat{\bm{n}}$ which exceeds that of the lower-density nematic and an intermittent smectic A phase by about an order of magnitude. Here we demonstrate that the unique transport behavior in the RN phase may be linked to a confinement-induced packing effect which causes the formation of supramolecular, string-like conformations. The strings consist of several individual molecules that are capable of travelling in the direction of $\hat{\bm{n}}$ as individual "trains" consisting of chains of molecular "cars". Individual trains run in parallel and may pass each other at sufficiently high pressures.Comment: 24 page

Advanced composite applications for sub-micron biologically derived microstructures

A major thrust of advanced material development is in the area of self-assembled ultra-fine particulate based composites (micro-composites). The application of biologically derived, self-assembled microstructures to form advanced composite materials is discussed. Hollow 0.5 micron diameter cylindrical shaped microcylinders self-assemble from diacetylenic lipids. These microstructures have a multiplicity of potential applications in the material sciences. Exploratory development is proceeding in application areas such as controlled release for drug delivery, wound repair, and biofouling as well as composites for electronic and magnetic applications, and high power microwave cathodes

Aircraft and avionic related research required to develop an effective high-speed runway exit system

Research was conducted to increase airport capacity by studying the feasibility of the longitudinal separation between aircraft sequences on final approach. The multidisciplinary factors which include the utility of high speed exits for efficient runway operations were described along with recommendations and highlights of these studies