Dipartimento di Matematica "Giuseppe Peano"

Lezione lagrangiana

On the ergodicity of the frame flow

Relatore:
Andrei Moroianu
CNRS Paris-Saclay

Data e luogo:
15 Maggio 2025 dalle 14.30 alle 15.30
in Aula A, Palazzo Campana, Via Carlo Alberto 10, Torino

Abstract  

 The geodesic flow of compact Riemannian manifolds with negative sectional curvature is hyperbolic, thus ergodic with respect to the Liouville measure on the unit tangent bundle. However, 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 or quaternion-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 1/4 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 Semmelmann (Stuttgart University).

Short Bio 

Andrei Moroianu graduated at the Ecole Normale Supérieure in Paris in 1995 and got his PhD from the Ecole Polytechnique in 1996. He works for the CNRS since 1997, and his research topic is differential geometry. Since 2017 he is based at the Paris-Saclay University. He was awarded the Peccot Lecture at the Collège de France in 1998 and he held lectures at the Ecole Polytechnique from 2004 to 2016. He is an author of over 90 research papers and two monographs and he currently leads a team of 20 researchers within a PNRR Grant of 1.2M€ funded by the European Union.