Work

On the ergodicity of the frame flow on even-dimensional manifolds

Year: 2024

Type: article

Abstract: It is known that the frame flow on a closed $n$ n -dimensional Riemannian manifold with negative sectional curvature is ergodic if $n$ n is odd and $n \neq 7$ n ≠ 7 . In this paper we study its ergodicity in the remaining cases. For $n$ n even and $n \neq 8, 134$ n ≠ 8 , 134 , we show that: if $n \equiv 2$ n ≡ 2 mod 4 or $n=4$ n = 4 , the frame flow is ergodic if the manifold is $\sim 0.3$ ∼ 0.3 -pinched, if $n \equiv 0$ n ≡ 0 mod 4, it is ergodic if the manifold is $\sim 0.6$ ∼ 0.6 -pinched. In the three dimensions $n=7,8,134$ n = 7 , 8 , 134 , the respective pinching bounds that we need in order to prove ergodicity are 0.4962..., 0.6212..., and 0.5788.... This is a significant improvement over the previously known results and a step forward towards solving a long-standing conjecture of Brin asserting that 0.25-pinched even-dimensional manifolds have an ergodic frame flow.

Source: Inventiones mathematicae

DOI: 10.1007/s00222-024-01297-7

Authors: Mihajlo Cekić, Thibault Lefeuvre, Andrei Moroianu, Uwe Semmelmann