We study asymptotically flat axially symmetric stationary solutions of the Einstein vacuum equations. These represent rotating black holes in equilibrium. The equations reduce outside the axis of symmetry to a harmonic map problem into the hyperbolic plane, with prescribed rates of blow-up for the map on the axis and at infinity as boundary conditions. We prove existence and uniqueness of solutions in the case of zero total angular momentum.

We extend the validity of Dain's angular-momentum inequality to maximal, asymptotically flat, initial data sets on a simply connected manifold with several asymptotically flat ends which are invariant under a U(1) action and which admit a twist potential.

ApoA-I is a uniquely flexible lipid-scavenging protein capable of incorporating phospholipids into stable particles. Here we report molecular dynamics simulations on a series of progressively smaller discoidal high density lipoprotein particles produced by incremental removal of palmitoyloleoylphosphatidylcholine via four different pathways. The starting model contained 160 palmitoyloleoylphosphatidylcholines and a belt of two antiparallel amphipathic helical lipid-associating domains of apolipoprotein (apo) A-I. The results are particularly compelling. After a few nanoseconds of molecular dynamics simulation, independent of the starting particle and method of size reduction, all simulated double belts of the four lipidated apoA-I particles have helical domains that impressively approximate the x-ray crystal structure of lipid-free apoA-I, particularly between residues 88 and 186. These results provide atomic resolution models for two of the particles produced by in vitro reconstitution of nascent high density lipoprotein particles. These particles, measuring 95 angstroms and 78 angstroms by nondenaturing gradient gel electrophoresis, correspond in composition and in size/shape (by negative stain electron microscopy) to the simulated particles with molar ratios of 100:2 and 50:2, respectively. The lipids of the 100:2 particle family form minimal surfaces at their monolayer-monolayer interface, whereas the 50:2 particle family displays a lipid pocket capable of binding a dynamic range of phospholipid molecules.

We prove that a broad subset of the space of asymptotically flat Riemannian metrics of nonnegative scalar curvature on R 3 is connected using a new method for prescribing scalar curvature that generalizes a method developed by Bartnik for quasi-spherical metrics.

We construct a time-symmetric asymptotically flat initial data set to the Einstein-Maxwell Equations which satisfies the inequality: m - 1/2(R + Q^2/R) < 0, where m is the total mass, R=sqrt(A/4) is the area radius of the outermost horizon and Q is the total charge. This yields a counter-example to a natural extension of the Penrose Inequality to charged black holes.Comment: Minor revision: some typos; author's address updated; bibliographical reference added; journal information: to appear in Comm. Math. Phy