We prove that the geodesic flow for the Weil-Petersson metric on the moduli space of Riemann surfaces is ergodic (and in fact Bernoulli) and has finite, positive metric entropy.

The explicit solutions of the Bogomolny equations for N vortices on a sphere of radius R2 > N are not known. In particular, this has prevented the use of the geodesic approximation to describe the low energy vortex dynamics. In this paper we introduce an approximate general solution of the equations, valid for R2 & N , which has many properties of the true solutions, including the same moduli s...

We describe a new proof of the complete integrability of the two related dynamical systems: the billiard inside the ellipsoid and the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the ellipsoid whose nonparameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperb...

In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (locally) a metric space is to a tree from a metric point of view. The study of Gromov hyperbolicity for geodesic metric spaces can be reduced to the study of graph hyperbolicity. Our main contribution in this note is a new characterization of hyperbolicity for graphs (and for complete geodesic metric ...

A geodesic metric space is δ-hyperbolic in the sense of Gromov if and only if the intersection of any two metric balls is almost a ball. In particular, R-trees can be characterised by the property that the intersection of any two metric balls is again a metric ball.

A novel scheme for texture segmentation is presented. Our algorithm is based on generalizing the intensity-based geodesic active contours model to the Gabor spatial-feature space of images. First, we apply the Gabor-Morlet transform to the image using self similar Gabor functions, and then implement the geodesic active snakes mechanism in this space. The spatial-feature space is represented, vi...

Elastic shape analysis on non-linear Riemannian manifolds provides an efficient and elegant way for simultaneous comparison and registration of non-rigid shapes. In such formulation, shapes become points on some high dimensional shape space. A geodesic between two points corresponds to the optimal deformation needed to register one shape onto another. The length of the geodesic provides a prope...