The Virasoro-Bott group endowed with the right-invariant L2metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.

Let X, d be a metric space and x, y ∈ X with l d x, y . A geodesic path from x to y is an isometry c : 0, l → X such that c 0 x and c l y. The image of a geodesic path is called a geodesic segment. A metric spaceX is a (uniquely) geodesic space if every two points ofX are joined by only one geodesic segment. A geodesic triangle x1, x2, x3 in a geodesic space X consists of three points x1, x2, x...

Providing each simplex in C(S) with the standard euclidean metric of side-length 1 equips the complex of curves with the structure of a geodesic metric space whose isometry group is just M̃g,m (except for the twice punctured torus). However, this metric space is not locally compact. Masur and Minsky [MM1] showed that nevertheless the geometry of C(S) can be understood quite explicitly. Namely, C...

Low-energy dynamics in the unit-charge sector of the CP 1 model on spherical space (space-time S × R) is treated in the approximation of geodesic motion on the moduli space of static solutions, a sixdimensional manifold with non-trivial topology and metric. The structure of the induced metric is restricted by consideration of the isometry group inherited from global symmetries of the full field...

Low-energy dynamics in the unit-charge sector of the CP 1 model on spherical space (space-time S 2 × R) is treated in the approximation of geodesic motion on the moduli space of static solutions, a six-dimensional manifold with non-trivial topology and metric. The structure of the induced metric is restricted by consideration of the isometry group inherited from global symmetries of the full fi...

We continue our investigation of the space of geodesic laminations on a surface, endowed with the Hausdorff topology. We determine the topology of this space for the once punctured torus and the 4–times punctured sphere. For these two surfaces, we also compute the Hausdorff dimension of the space of geodesic laminations, when it is endowed with the natural metric which, for small distances, is ...

The notion of Gromov hyperbolicity was introduced by Gromov in the setting of geometric group theory [G1], [G2], but has played an increasing role in analysis on general metric spaces [BHK], [BS], [BBo], [BBu], and extendability of Lipschitz mappings [L]. In this theory, it is often additionally assumed that the hyperbolic metric space is proper and geodesic (meaning that closed balls are compa...

We show that both Teichmüller space (with the Teichmüller metric) and the mapping class group (with a word metric) have geodesic divergence that is intermediate between the linear rate of flat spaces and the exponential rate of hyperbolic spaces. For every two geodesic rays in Teichmüller space, we find that their divergence is at most quadratic. Furthermore, this estimate is shown to be sharp ...

Given two pointed Gromov hyperbolic metric spaces (Xi, di, zi), i = 1, 2, and ∆ ∈ R+0 , we present a construction method, which yields another Gromov hyperbolic metric space Y∆ = Y∆((X1, d1, z1), (X2, d2, z2)). Moreover, it is shown that once (Xi, di) is roughly geodesic, i = 1, 2, then there exists a ∆′ ≥ 0 such that Y∆ also is roughly geodesic for all ∆ ≥ ∆ ′.

We show that if a domain Ω in a geodesic metric space is quasimöbius to a uniform domain in some metric space, then Ω is also uniform. Mathematics Subject Classification (2000). 30C65