Non-rigidity of spherical inversive distance circle packings

We give a counterexample of Bowers-Stephenson's conjecture in the spherical case: spherical inversive distance circle packings are not determined by their inversive distances.Comment: 6 pages, one pictur

Minimal surfaces and particles in 3-manifolds

We use minimal (or CMC) surfaces to describe 3-dimensional hyperbolic, anti-de Sitter, de Sitter or Minkowski manifolds. We consider whether these manifolds admit ``nice'' foliations and explicit metrics, and whether the space of these metrics has a simple description in terms of Teichm\"uller theory. In the hyperbolic settings both questions have positive answers for a certain subset of the quasi-Fuchsian manifolds: those containing a closed surface with principal curvatures at most 1. We show that this subset is parameterized by an open domain of the cotangent bundle of Teichm\"uller space. These results are extended to ``quasi-Fuchsian'' manifolds with conical singularities along infinite lines, known in the physics literature as ``massive, spin-less particles''. Things work better for globally hyperbolic anti-de Sitter manifolds: the parameterization by the cotangent of Teichm\"uller space works for all manifolds. There is another description of this moduli space as the product two copies of Teichm\"uller space due to Mess. Using the maximal surface description, we propose a new parameterization by two copies of Teichm\"uller space, alternative to that of Mess, and extend all the results to manifolds with conical singularities along time-like lines. Similar results are obtained for de Sitter or Minkowski manifolds. Finally, for all four settings, we show that the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichm\"uller space is the same as the 3-dimensional gravity one.Comment: 53 pages, no figure. v2: typos corrected and refs adde

On the total mean curvature of non-rigid surfaces

Using Green's theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field and obtain the following well-known theorem as an immediate consequence: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex.Comment: 4 page

Collisions of particles in locally AdS spacetimes I. Local description and global examples

We investigate 3-dimensional globally hyperbolic AdS manifolds containing "particles", i.e., cone singularities along a graph $\Gamma$. We impose physically relevant conditions on the cone singularities, e.g. positivity of mass (angle less than $2\pi$ on time-like singular segments). We construct examples of such manifolds, describe the cone singularities that can arise and the way they can interact (the local geometry near the vertices of $\Gamma$). We then adapt to this setting some notions like global hyperbolicity which are natural for Lorentz manifolds, and construct some examples of globally hyperbolic AdS manifolds with interacting particles.Comment: This is a rewritten version of the first part of arxiv:0905.1823. That preprint was too long and contained two types of results, so we sliced it in two. This is the first part. Some sections have been completely rewritten so as to be more readable, at the cost of slightly less general statements. Others parts have been notably improved to increase readabilit

Non-fermi-liquid single particle lineshape of the quasi-one-dimensional non-CDW metal Li_{0.9}Mo_{6}O_{17} : comparison to the Luttinger liquid

We report the detailed non-Fermi liquid (NFL) lineshape of the dispersing excitation which defines the Fermi surface (FS) for quasi-one-dimensional Li_{0.9}Mo_{6}O_{17}. The properties of Li_{0.9}Mo_{6}O_{17} strongly suggest that the NFL behavior has a purely electronic origin. Relative to the theoretical Luttinger liquid lineshape, we identify significant similarities, but also important differences.Comment: 5 pages, 3 eps figure

Non-Fermi liquid angle resolved photoemission lineshapes of Li0.9Mo6O17

A recent letter by Xue et al. (PRL v.83, 1235 ('99)) reports a Fermi-Liquid (FL) angle resolved photoemission (ARPES) lineshape for quasi one-dimensional Li0.9Mo6O17, contradicting our report (PRL v.82, 2540 ('99)) of a non-FL lineshape in this material. Xue et al. attributed the difference to the improved angle resolution. In this comment, we point out that this reasoning is flawed. Rather, we find that their data have fundamental differences from other ARPES results and also band theory.Comment: To be published as a PRL Commen

Two-dimensional Hubbard-Holstein bipolaron

We present a diagrammatic Monte Carlo study of the properties of the Hubbard-Holstein bipolaron on a two-dimensional square lattice. With a small Coulomb repulsion, U, and with increasing electron-phonon interaction, and when reaching a value about two times smaller than the one corresponding to the transition of light polaron to heavy polaron, the system suffers a sharp transition from a state formed by two weakly bound light polarons to a heavy, strongly bound on-site bipolaron. Aside from this rather conventional bipolaron a new bipolaron state is found for large U at intermediate and large electron-phonon coupling, corresponding to two polarons bound on nearest-neighbor sites. We discuss both the properties of the different bipolaron states and the transition from one state to another. We present a phase diagram in parameter space defined by the electron-phonon coupling and U. Our numerical method does not use any artificial approximation and can be easily modified to other bipolaron models with longer range electron-phonon and/or electron-electron interaction.Comment: 14 pages, 12 figure

Coherent amplitudon generation in K_0.3MoO_3 through ultrafast inter-band quasi particle decay

The charge density wave system K_0.3MoO_3 has been studied using variable energy pump-probe spectroscopy, ellipsometry, and inelastic light scattering. The observed transient reflectivity response exhibits quite a complex behavior, containing contributions due to quasi particle excitations, coherent amplitudons and phonons, and heating effects. The generation of coherent amplitudons is discussed in terms of relaxation of photo-excited quasi particles, and is found to be resonant with the interband plasmon frequency. Two additional coherent excitations observed in the transients are assigned to zone-folding modes of the charge density wave state

Fuchsian convex bodies: basics of Brunn--Minkowski theory

The hyperbolic space \H^d can be defined as a pseudo-sphere in the $(d+1)$ Minkowski space-time. In this paper, a Fuchsian group $\Gamma$ is a group of linear isometries of the Minkowski space such that \H^d/\Gamma is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be associated to each Fuchsian convex body, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn--Minkowski theory. For example, support functions can be defined, as functions on a compact hyperbolic manifold instead of the sphere. The main result is the convexity of the associated volume (it is log concave in the classical setting). This implies analogs of Alexandrov--Fenchel and Brunn--Minkowski inequalities. Here the inequalities are reversed