12-01【Andrei Moroianu】五教5305 几何拓扑及高阶Teichmuller研讨班系列报告之二十二


时间: 2023年12月1日, 上午10:00


报告人:Andrei Moroianu, CNRS

题目:On the ergodicity of the frame flow

摘要:The geodesic flow of compact Riemannian manifolds with negative sectional curvature is ergodic with respect to the Liouville measure on the unit tangent bundle. Its natural lift to the orthonormal frame bundle, called the frame flow, is only partially hyperbolic, and is not always ergodic, e.g. when the manifold is Kähler. In the 80's, Brin and Gromov proved the ergodicity of the frame flow when the dimension $n$ of the manifold is odd and different from 7, and conjectured that ergodicity should hold whenever the sectional curvature is at least 0,25 pinched. Some partial results were obtained by Brin-Karcher ('84) and Burns-Pollicott ('03), under quite strong pinching assumptions of the order of 0,86 and 0,98. In this talk I will report on some recent developments towards the Brin-Gromov conjecture, where the required pinching is approximately 0,27 - 0,29 for $n=4k+2$ and 0,57 - 0,6 for $n=4k$. The talk is based on joint work with Mihajlo Cekic (ETH, Zürich), Thibault Lefeuvre (Sorbonne Université) and Uwe Semmelman (Stuttgart University).