On the ergodicity of unitary frame flows on Kähler manifolds

Abstract

Abstract Let $(M,g,J)$ be a closed Kähler manifold with negative sectional curvature and complex dimension $m := \dim _{\mathbb {C}} M \geq 2$ . In this article, we study the unitary frame flow , that is, the restriction of the frame flow to the principal $\mathrm {U}(m)$ -bundle $F_{\mathbb {C}}M$ of unitary frames. We show that if $m \geq 6$ is even and $m \neq 28$ , there exists $\unicode{x3bb} (m) \in (0, 1)$ such that if $(M, g)$ has negative $\unicode{x3bb} (m)$ -pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants $\unicode{x3bb} (m)$ satisfy $\unicode{x3bb} (6) = 0.9330...$ , $\lim _{m \to +\infty } \unicode{x3bb} (m) = {11}/{12} = 0.9166...$ , and $m \mapsto \unicode{x3bb} (m)$ is decreasing. This extends to the even-dimensional case the results of Brin and Gromov [On the ergodicity of frame flows. Invent. Math. 60 (1) (1980), 1–7] who proved ergodicity of the unitary frame flow on negatively curved compact Kähler manifolds of odd complex dimension.