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

TL;DR

This paper investigates the conditions under which the frame flow on even-dimensional negatively curved manifolds is ergodic, providing improved pinching bounds and advancing towards a long-standing conjecture.

Contribution

It extends ergodicity results to more cases of even-dimensional manifolds with specific pinching conditions, improving previous bounds and addressing a major conjecture.

Findings

Ergodicity proven for certain even dimensions with specific pinching constants

Improved pinching bounds for ergodicity in dimensions 7, 8, and 134

Progress towards Brin's conjecture on 0.25-pinched manifolds

Abstract

It is known that the frame flow on a closed $n$-dimensional Riemannian manifold with negative sectional curvature is ergodic if $n$ is odd and $n \neq 7$. In this paper we study its ergodicity in the remaining cases. For $n$ even and $n \neq 8, 134$, we show that: if $n \equiv 2$ mod $4$ or $n=4$, the frame flow is ergodic if the manifold is $\sim 0.3$-pinched, if $n \equiv 0$ mod $4$, it is ergodic if the manifold is $\sim 0.6$-pinched. In the three dimensions $n=7,8,134$, 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.