Extreme points in non-positive curvature

Martin Points on Open Manifolds of Non-positive Curvature

The Martin boundary of a Cartan-Hadamard manifold describes a fine geometric structure at infinity, which is a sub-space of positive harmonic functions. We describe conditions which ensure that some points of the sphere at infinity belong to the Martin boundary as well. In the case of the universal cover of a compact manifold with Ballmann rank one, we show that Martin points are generic and of...

Non-linear ergodic theorems in complete non-positive curvature metric spaces

Hadamard (or complete $CAT(0)$) spaces are complete, non-positive curvature, metric spaces. Here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. Our results extend the standard non-linear ergodic theorems for non-expansive maps on real Hilbert spaces, to non-expansive maps on Ha...

of non-positive Gaussian curvature

On the Number of Geodesic Segments Connecting Two Points on Manifolds of Non-positive Curvature

1. Introduction Let M be a compact manifold Riemannian manifold of dimension n ≥ 2, with a metric of sectional curvature bounded above by χ ≤ 0 (non-positive curvature). In this paper we prove that in the case of negative curvature (χ < 0) on such manifolds there exist pairs of points connected by at least 2n + 1 geometrically distinct geodesic segments (i.e. length minimizing). A class of poin...

Dehn Functions and Non-Positive Curvature