where χ(S) is the Euler characteristic of the surface, B r denotes the geodesic r-ball in Hn(b) and Vol(S 2∩Bb,n r ) Vol(B r ) is the volume growth of the domains S2 ∩B r . A natural question arises in this context: can we prove the finiteness of the topology of a not necessarily minimal surface in a Cartan–Hadamard manifold and, moreover, establish a Chern–Osserman-type inequality for its Eule...

The interior of every compact contractible PL n-manifold (n 5) supports a complete geodesic metric of strictly negative curvature. This provides a new family of simple examples illustrating the negative answer to a question of M. Gromov which asks whether metrically convex geodesic spaces which are topological manifolds must be homeomorphic to Euclidean spaces. The rst examples verifying the ne...

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...

We study the equation ? ? g w + = ? ? ( ? ) f on a d -dimensional homogeneous Cartan-Hadamard Manifold M with ? 3 . Without using theory of topological indices, we prove existence infinitely many solutions for class nonlinearities which have an oscillating behavior either at zero or infinity.

In this paper, the Cauchy problem for a Friedrichs system on globally hyperbolic manifold with timelike boundary is investigated. By imposing admissible conditions, existence and uniqueness of strong solutions are shown. Furthermore, if hyperbolic, proved to be well-posed in sense Hadamard. Finally, examples systems conditions provided.

The purpose of this article was to establish manifold versions existence theorems for generalized vector quasi-equilibrium problems in locally compact and σ-compact spaces without any continuity assumption. fixed-point theorem a product Hadamard is the key focus our discussion. We further applied saddle point minimax problems.

We present an extension of the proximal point method with Bregman distances to solve Variational Inequality Problems (VIP) on Hadamard manifolds (simply connected finite dimensional Riemannian manifold with nonpositive sectional curvature). Under some natural assumption, as for example, the existence of solutions of the (VIP) and the monotonicity of the multivalued vector field, we prove that t...

We present a new self-contained and rigorous proof of the smoothness of invariant fiber bundles for dynamic equations on measure chains or time scales. Here, an invariant fiber bundle is the generalization of an invariant manifold to the nonautonomous case. Our main result generalizes the “Hadamard-Perron theorem” to the time-dependent, infinitedimensional, noninvertible, and parameter-dependen...

I. Introduction-a quick historical survey of geodesic flows on negatively curved spaces. II. Preliminaries on Riemannian manifolds A. Riemannian metric and Riemannian volume element B. Levi Civita connection and covariant differentiation along curves C. Parallel translation of vectors along curves D. Curvature E. Geodesics and geodesic flow F. Riemannian exponential map and Jacobi vector fields...