202 research outputs found
Hyperideal circle patterns
A ``hyperideal circle pattern'' in is a finite family of oriented
circles, similar to the ``usual'' circle patterns but such that the closed
disks bounded by the circles do not cover the whole sphere. Hyperideal circle
patterns are directly related to hyperideal hyperbolic polyhedra, and also to
circle packings.
To each hyperideal circle pattern, one can associate an incidence graph and a
set of intersection angles. We characterize the possible incidence graphs and
intersection angles of hyperideal circle patterns in the sphere, the torus, and
in higher genus surfaces. It is a consequence of a more general result,
describing the hyperideal circle patterns in the boundaries of geometrically
finite hyperbolic 3-manifolds (for the corresponding \C P^1-structures). This
more general statement is obtained as a consequence of a theorem of Otal
\cite{otal,bonahon-otal} on the pleating laminations of the convex cores of
geometrically finite hyperbolic manifolds.Comment: 11 pages, 2 figures. Updated versions will be posted on
http://picard.ups-tlse.fr/~schlenker Revised version: some corrections,
better proof, added reference
Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian manifolds
The boundary at infinity of a quasifuchsian hyperbolic manifold is equiped
with a holomorphic quadratic differential. Its horizontal measured foliation
can be interpreted as the natural analog of the measured bending lamination
on the boundary of the convex core. This analogy leads to a number of
questions. We provide a variation formula for the renormalized volume in terms
of the extremal length \ext(f) of , and an upper bound on \ext(f).
We then describe two extensions of the holomorphic quadratic differential at
infinity, both valid in higher dimensions. One is in terms of
Poincar\'e-Einstein metrics, the other (specifically for conformally flat
structures) of the second fundamental form of a hypersurface in a "constant
curvature" space with a degenerate metric, interpreted as the space of
horospheres in hyperbolic space. This clarifies a relation between linear
Weingarten surfaces in hyperbolic manifolds and Monge-Amp\`ere equations.Comment: Notes aiming at clarifying the relations between different points of
view and introducing one new notion, no real result. Not intended to be
submitted at this poin
- …