Entropy-expansiveness of Geodesic Flows on Closed Manifolds without Conjugate Points

In this article,we consider the entropy-expansiveness of geodesic flows on closed Riemannian manifolds without conjugate points.We prove that,if the manifold has no focal points,or if the manifold is bounded asymptote,then the geodesic flow is entropy-expansive.Moreover,for the compact oriented surfaces without conjugate points,we prove that the geodesic flows are entropy-expansive.We also give an estimation of distance between two positively asymptotic geodesics of an uniform visibility manifold.

Ergodic Measures of Geodesic Flows on Compact Lie Groups

Let Ψ be the geodesic flow associated with a two-sided invariant metric on a compact Lie group. In this paper, we prove that every ergodic measure μ of Ψ is supported on the unit tangent bundle of a flat torus. As an application, all Lyapunov exponents of μ are zero hence μ is not hyperbolic. Our underlying manifolds have nonnegative curvature （possibly strictly positive on some sections）, whereas in contrast, all geodesic flows related to negative curvature are Anosov hence every ergodic measure is hyperbolic.

Geodesic flows on path spaces

On the path space over a compact Riemannian manifold, the global existence and the global uniqueness of the quasi-invariant geodesic flows with respect to a negative Markov connection are obtained in this paper. The results answer affirmatively a left problem of Li.

Self-intersecting Geodesics and Entropy of the Geodesic Flow

Given a symmetric Finsler metric on T^2 whose geodesic flow has zero topological entropy, we show that the lift in the universal covering R^2 →T^2 of any closed geodesic on T^2 must be an embedded curve in R^2.

A Spectral Segmentation Method for Large Meshes

Mesh segmentation is a fundamental and critical task in mesh processing,and it has been studied extensively in computer graphics and geometric modeling communities.However,current methods are not well suited for segmenting large meshes which are now common in many applications.This paper proposes a new spectral segmentation method specifically designed for large meshes inspired by multi-resolution representations.Building on edge collapse operators and progressive mesh representations,we first devise a feature-aware simplification algorithm that can generate a coarse mesh which keeps the same topology as the input mesh and preserves as many features of the input mesh as possible.Then,using the spectral segmentation method proposed in Tong et al.(IEEE Trans Vis Comput Graph 26(4):1807–1820,2020),we perform partition on the coarse mesh to obtain a coarse segmentation which mimics closely the desired segmentation of the input mesh.By reversing the simplification process through vertex split operators,we present a fast algorithm which maps the coarse segmentation to the input mesh and therefore obtain an initial segmentation of the input mesh.Finally,to smooth some jaggy boundaries between adjacent parts of the initial segmentation or align with the desired boundaries,we propose an efficient method to evolve those boundaries driven by geodesic curvature flows.As demonstrated by experimental results on a variety of large meshes,our method outperforms the state-of-the-art segmentation method in terms of not only speed but also usability.

The Uniqueness of Knieper Measure on Non-compact Rank 1 Manifolds of Non-positive Curvature

We study the Knieper measures of the geodesic flows on non-compact rank 1 manifolds of non-positive curvature.We construct the Busemann density on the ideal boundary,and prove that if there is a Knieper measure on T^(1)M with finite total mass,then the Knieper measure is unique,up to a scalar multiple.Our result partially extends Paulin-Pollicott-Shapira’s work on the uniqueness of finite Gibbs measure of geodesic flows on negatively curved non-compact manifolds to non-compact manifolds of non-positive curvature.

Asymptotic Formulae for Brillouin Index on Riemannian Manifolds

We establish several asymptotic formulae for Brillouin index on fiat tori. As an application of these formulae it is proved that the topological entropy of a geodesic flow on a fiat torus is zero.