342 research outputs found
Slicing, skinning, and grafting
We prove that a Bers slice is never algebraic, meaning that its Zariski
closure in the character variety has strictly larger dimension. A corollary is
that skinning maps are never constant.
The proof uses grafting and the theory of complex projective structures.Comment: 11 pages, 1 figure, to appear in American Journal of Mathematic
A survey of some arithmetic applications of ergodic theory in negative curvature
This paper is a survey of some arithmetic applications of techniques in the
geometry and ergodic theory of negatively curved Riemannian manifolds, focusing
on the joint works of the authors. We describe Diophantine approximation
results of real numbers by quadratic irrational ones, and we discuss various
results on the equidistribution in , and in the
Heisenberg groups of arithmetically defined points. We explain how these
results are consequences of equidistribution and counting properties of common
perpendiculars between locally convex subsets in negatively curved orbifolds,
proven using dynamical and ergodic properties of their geodesic flows. This
exposition is based on lectures at the conference "Chaire Jean Morlet:
G\'eom\'etrie et syst\`emes dynamiques", at the CIRM, Luminy, 2014. We thank B.
Hasselblatt for his strong encouragements to write this survey.Comment: 31 pages, 15 figure
The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores
In this paper, we use the Weil-Petersson gradient flow for renormalized
volume to study the space of convex cocompact hyperbolic structures
on the relatively acylindrical 3-manifold . Among the cases of interest
are the deformation space of an acylindrical manifold and the Bers slice of
quasi-Fuchsian space associated to a fixed surface. To treat the possibility of
degeneration along flow-lines to peripherally cusped structures, we introduce a
surgery procedure to yield a surgered gradient flow that limits to the unique
structure with totally geodesic convex core
boundary facing . Analyzing the geometry of structures along a flow line, we
show that if is the renormalized volume of , then
is bounded below by a linear function of the
Weil-Petersson distance ,
with constants depending only on the topology of . The surgered flow gives a
unified approach to a number of problems in the study of hyperbolic
3-manifolds, providing new proofs and generalizations of well-known theorems
such as Storm's result that has minimal volume for
acylindrical and the second author's result comparing convex core volume and
Weil-Petersson distance for quasifuchsian manifolds
Skeleton Driven Non-rigid Motion Tracking and 3D Reconstruction
This paper presents a method which can track and 3D reconstruct the non-rigid
surface motion of human performance using a moving RGB-D camera. 3D
reconstruction of marker-less human performance is a challenging problem due to
the large range of articulated motions and considerable non-rigid deformations.
Current approaches use local optimization for tracking. These methods need many
iterations to converge and may get stuck in local minima during sudden
articulated movements. We propose a puppet model-based tracking approach using
skeleton prior, which provides a better initialization for tracking articulated
movements. The proposed approach uses an aligned puppet model to estimate
correct correspondences for human performance capture. We also contribute a
synthetic dataset which provides ground truth locations for frame-by-frame
geometry and skeleton joints of human subjects. Experimental results show that
our approach is more robust when faced with sudden articulated motions, and
provides better 3D reconstruction compared to the existing state-of-the-art
approaches.Comment: Accepted in DICTA 201
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
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