On the ergodicity of unitary frame flows on Kähler manifolds (Q6624539)

The ergodicity (with respect to the Liouville measure) of the geodesic flow on the unit tangent bundle on a negatively curved compact Riemannian manifold has been well-known since the early work of \textit{E. Hopf} [Trans. Am. Math. Soc. 39, 299--314 (1936; JFM 62.0995.01)] and its generalization by \textit{D. V. Anosov} [Sov. Math., Dokl. 4, 1153--1156 (1963; Zbl 0135.40402); translation from Dokl. Akad. Nauk SSSR 151, 1250--1252 (1963)]. For extensions of the geodesic flow to principal bundles the flow is not uniformly hyperbolic but is only partially hyperbolic, making all such questions more subtle. In this paper ergodicity of the unitary frame flow obtained as the restriction of the frame flow of a closed Kähler manifold with negative sectional curvature and complex dimension \(m\geqslant 2\) to the principal \({\mathrm{U}}(m)\)-bundle of unitary frames is studied. This flow preserves a naturally defined smooth measure induced by the Liouville measure and Haar measure on \({\mathrm{U}}(m-1)\). For \(m=2\) or \(m\) odd, \textit{M. Brin} and \textit{M. Gromov} [Invent. Math. 60, 1--7 (1980; Zbl 0445.58023)] established ergodicity. The main result here is to establish ergodicity for even \(m\geqslant 6\) and \(m\neq28\) under a pinching hypothesis for the sectional curvature.