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...

We consider the space of geodesic laminations on a surface, endowed with the Hausdorff metric dH and with a variation of this metric called the dlog metric. We compute and/or estimate the Hausdorff dimensions of these two metrics. We also relate these two metrics to another metric which is combinatorially defined in terms of train tracks. AMS Classification numbers Primary: 57M99

In this paper we study isotropic Lagrangian submanifolds , in complex space forms . It is shown that they are either totally geodesic or minimal in the complex projective space , if . When , they are either totally geodesic or minimal in . We also give a classification of semi-parallel Lagrangian H-umbilical submanifolds.

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish non branching geodesic space. We show that we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce an assumption on the transport problem π which implies that the conditional probabilities of the first marginal on each geodesic are continuous. It is known that...

and Applied Analysis 3 by 1.5 to a common fixed point of a countable infinite family of nonexpansive mappings in convex metric spaces and CAT 0 spaces under certain suitable conditions. 2. Preliminaries We recall some definitions and useful lemmas used in the main results. Lemma 2.1 see 9, 10 . Let X, d,W be a convex metric space. For each x, y ∈ X and λ, λ1, λ2 ∈ 0, 1 , we have the following. ...

After a brief introduction on gradient flows in metric spaces and on geodesically convex functionals, we give an account of the proof (following the outline of [3, 7]) of the existence and uniqueness of the gradient flow of the Entropy in the space of Borel probability measures over a compact geodesic metric space with Ricci curvature bounded from below. Preprint SISSA 17/2014/MATE

The Lagrangian for the motion of n well-separated BPS monopoles is calculated, by treating the monopoles as point particles with magnetic, electric and scalar charges. It can be reinterpreted as the Lagrangian for geodesic motion on the asymptotic region of the n-monopole moduli space, thereby determining the asymptotic metric on the moduli space. The metric is hyperkähler, and is an explicit e...

Every point in Teichmüller space is a hyperbolic metric on a given Riemann surface, therefore, a Weil-Petersson geodesic in Teichmüller space can be viewed as a 3-manifold. We investigate the sectional curvatures of this 3-manifold, with a natural metric. We obtain explicit formulas for the curvature tensors of this metric, and show that the “average”s of them are zero, and hence the geometry o...

An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the study of ramified optimal transportation between probability measures from Euclidean spaces to a geodesic ...

Main results. Let Tg,n andMg,n denote the Teichmüller and moduli space respectively of genus g Riemann surfaces with n marked points. The Teichmüller metric on these spaces is a natural Finsler metric that quantifies the failure of two different Riemann surfaces to be conformally equivalent. It is equal to the Kobayashi metric [Roy74], and hence reflects the intrinsic complex geometry of these ...