Random walks in negative curvature and boundary representations

变负曲率几何无限曲面上的极限圆轨道与测地轨道

Horocyclic and geodesic orbits on geometrically infinite surfaces of variable negative curvature

本文研究变负曲率几何无限曲面上，当测地射线几乎极小化且单射半径有限时，单位切丛中向量极限圆轨道的行为。

Flexibility of entropies for surfaces of negative curvature

We consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a Riemannian metric $g$ of negative curvature with fixed total area. The second author has shown that the topological entropy of geodesic flow for $g$ is greater than or equal to the topological entropy for the metric of constant negative curvature on $M$ with the same total area which is greater than or equal to the metric entropy with respect to the Liouville measure of geodesic flow for $g$. Equality holds only in the case of constant negative curvature. We prove that those are the only restrictions on the values of t…

Einstein manifolds of non-negative sectional curvature and entropy

We find obstructions to the existence of Einstein metrics of non-negative sectional curvature on a smooth closed simply connected manifold of any dimension. The results are achieved by combining the classical Morse theory of the loop space with a new upper bound for the topological entropy of the geodesic flow in terms of the curvature tensor.

Hamiltonian systems of negative curvature are hyperbolic

The {\it curvature} and the {\it reduced curvature} are basic differential invariants of the pair: (Hamiltonian system, Lagrange distribution) on the symplectic manifold. We show that negativity of the curvature implies that any bounded semi-trajectory of the Hamiltonian system tends to a hyperbolic equilibrium, while negativity of the reduced curvature implies the hyperbolicity of any compact invariant set of the Hamiltonian flow restricted to a prescribed energy level. Last statement generalizes a well-known property of the geodesic flows of Riemannian manifolds with negative sectional curva…

Central limit theorem of Brownian motions in pinched negative curvature

We prove the central limit theorem of random variables induced by distances to Brownian paths and Green functions on the universal cover of Riemannian manifolds of finite volume with pinched negative curvature. We further provide some ergodic properties of Brownian motions and an application of the central limit theorem to the dynamics of geodesic flows in pinched negative curvature.

负曲率黎曼流形上的定量闭引理与伙伴轨道

A quantitative closing Lemma and Partner orbits on Riemannian manifolds with negative curvature

该研究证明负截面曲率流形的定量闭引理，并将其应用于自交闭测地线的伙伴及伪伙伴轨道研究，为相关动力系统与几何问题提供定量分析工具。