182 research outputs found
An extension of the Maskit slice for 4-dimensional Kleinian groups
Let be a 3-dimensional Kleinian punctured torus group with ccidental
parabolic transformations. The deformation space of in the group of
M\"{o}bius transformations on the 2-sphere is well-known as the Maskit slice of
punctured torus groups. In this paper, we study deformations of
in the group of M\"{o}bius transformations on the 3-sphere such that
does not contain screw parabolic transformations. We will show that
the space of the deformations is realized as a domain of 3-space
, which contains the Maskit slice of punctured torus groups as a
slice through a plane. Furthermore, we will show that the space also contains
the Maskit slice of fourth-punctured sphere groups as a slice through another
plane. Some of another slices of the space will be also studied.Comment: 34 pages, 11 figures. v3: The title is changed and some typo are
fixed. To appear in Conform. Geom. dyn. The paper including more clear
figures can be downloaded from
http://www.math.nagoya-u.ac.jp/~itoken/index.htm
Black Hole Thermodynamics and Riemann Surfaces
We use the analytic continuation procedure proposed in our earlier works to
study the thermodynamics of black holes in 2+1 dimensions. A general black hole
in 2+1 dimensions has g handles hidden behind h horizons. The result of the
analytic continuation is a hyperbolic 3-manifold having the topology of a
handlebody. The boundary of this handlebody is a compact Riemann surface of
genus G=2g+h-1. Conformal moduli of this surface encode in a simple way the
physical characteristics of the black hole. The moduli space of black holes of
a given type (g,h) is then the Schottky space at genus G. The (logarithm of
the) thermodynamic partition function of the hole is the Kaehler potential for
the Weil-Peterson metric on the Schottky space. Bekenstein bound on the black
hole entropy leads us to conjecture a new strong bound on this Kaehler
potential.Comment: 17+1 pages, 9 figure
The classification of punctured-torus groups
Thurston's ending lamination conjecture proposes that a finitely generated
Kleinian group is uniquely determined (up to isometry) by the topology of its
quotient and a list of invariants that describe the asymptotic geometry of its
ends. We present a proof of this conjecture for punctured-torus groups. These
are free two-generator Kleinian groups with parabolic commutator, which should
be thought of as representations of the fundamental group of a punctured torus.
As a consequence we verify the conjectural topological description of the
deformation space of punctured-torus groups (including Bers' conjecture that
the quasi-Fuchsian groups are dense in this space) and prove a rigidity
theorem: two punctured-torus groups are quasi-conformally conjugate if and only
if they are topologically conjugate.Comment: 67 pages, published versio
Exact Results for the BTZ Black Hole
In this review, we summarize exact results for the three-dimensional BTZ
black hole. We use rigorous mathematical results to clarify the general
structure and properties of this black hole spacetime and its microscopic
description. In particular, we study the formation of the black hole by point
particle collisions, leading to an exact analytic determination of the Choptuik
scaling parameter. We also show that a `No Hair Theorem' follows immediately
from a mathematical theorem of hyperbolic geometry, due to Sullivan. A
microscopic understanding of the Bekenstein-Hawking entropy, and decay rate for
massless scalars, is shown to follow from standard results of conformal field
theory.Comment: 24 pages, Latex, Review article to appear in Int. J. Mod. Phys. D, v2
additional reference
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