16,043 research outputs found
Counting arcs in negative curvature
Let M be a complete Riemannian manifold with negative curvature, and let C_-,
C_+ be two properly immersed closed convex subsets of M. We survey the
asymptotic behaviour of the number of common perpendiculars of length at most s
from C_- to C_+, giving error terms and counting with weights, starting from
the work of Huber, Herrmann, Margulis and ending with the works of the authors.
We describe the relationship with counting problems in circle packings of
Kontorovich, Oh, Shah. We survey the tools used to obtain the precise
asymptotics (Bowen-Margulis and Gibbs measures, skinning measures). We describe
several arithmetic applications, in particular the ones by the authors on the
asymptotics of the number of representations of integers by binary quadratic,
Hermitian or Hamiltonian forms.Comment: Revised version, 44 page
Closed geodesics and holonomies for Kleinian manifolds
For a rank one Lie group G and a Zariski dense and geometrically finite
subgroup of G, we establish equidistribution of holonomy classes about
closed geodesics for the associated locally symmetric space. Our result is
given in a quantitative form for real hyperbolic geometrically finite manifolds
whose critical exponents are big enough. In the case when G=PSL(2, C), our
results can be interpreted as the equidistribution of eigenvalues of
in the complex plane.
When is a lattice, this result was proved by Sarnak and Wakayama in
1999.Comment: 28 pages, Minor corrections in the main term of the effective
versions of Theorem 1.2, 1.3 and 5.1 are made from the printed version
(GAFA,Vol 24 (2014) 1608-1636
Bulk-deformed potentials for toric Fano surfaces, wall-crossing and period
We provide an inductive algorithm to compute the bulk-deformed potentials for
toric Fano surfaces via wall-crossing techniques and a tropical-holomorphic
correspondence theorem for holomorphic discs. As an application of the
correspondence theorem, we also prove a big quantum period theorem for toric
Fano surfaces which relates the log descendant Gromov-Witten invariants with
the oscillatory integrals of the bulk-deformed potentials.Comment: 44 pages, 9 figures, comments are welcom
Entropy of random coverings and 4D quantum gravity
We discuss the counting of minimal geodesic ball coverings of -dimensional
riemannian manifolds of bounded geometry, fixed Euler characteristic and
Reidemeister torsion in a given representation of the fundamental group. This
counting bears relevance to the analysis of the continuum limit of discrete
models of quantum gravity. We establish the conditions under which the number
of coverings grows exponentially with the volume, thus allowing for the search
of a continuum limit of the corresponding discretized models. The resulting
entropy estimates depend on representations of the fundamental group of the
manifold through the corresponding Reidemeister torsion. We discuss the sum
over inequivalent representations both in the two-dimensional and in the
four-dimensional case. Explicit entropy functions as well as significant bounds
on the associated critical exponents are obtained in both cases.Comment: 54 pages, latex, no figure
On the arithmetic of crossratios and generalised Mertens' formulas
We develop the relation between hyperbolic geometry and arithmetic
equidistribution problems that arises from the action of arithmetic groups on
real hyperbolic spaces, especially in dimension up to 5. We prove
generalisations of Mertens' formula for quadratic imaginary number fields and
definite quaternion algebras over the rational numbers, counting results of
quadratic irrationals with respect to two different natural complexities, and
counting results of representations of (algebraic) integers by binary
quadratic, Hermitian and Hamiltonian forms with error bounds. For each such
statement, we prove an equidistribution result of the corresponding
arithmetically defined points. Furthermore, we study the asymptotic properties
of crossratios of such points, and expand Pollicott's recent results on the
Schottky-Klein prime functions.Comment: 44 page
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