On the ergodicity of the frame flow on even-dimensional manifolds (Q6647740)

This is a very original paper which address a longstanding problem of establishing ergodicity of frame flows in even dimensional negatively curved manifolds. Brin and coauthors studied this problem extensively and it was understood that when the manifold is odd-dimensional (except for \(n=7\)) ergodicity is always the case. However, for Kähler manifolds, ergodicity fails, and so it was conjectured by Brin that as long as the negative curvature is more than 1/4-pinched (which excludes Kähler counterexamples) then ergodicity should hold. This paper makes a substantial contribution to this conjecture, going from original bounds very close to \(1\)-pinching all the way to values very close to \(1/4\) (in the case of \(n=4\ell + 2\)) and close to \(1/2\) (in the case \(n=4\ell\)). Besides the interest of the results themselves, attention should be paid to the innovative techniques, that involve ideas from microlocal analysis to study both the ergodicity but also what in partially hyperbolic dynamics is called the accesibility problem.