On the ergodicity of unitary frame flows on K\"ahler manifolds

Abstract

Let $(M,g,J)$ be a closed K\"ahler 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 $\lambda(m) \in (0, 1)$ such that if $(M, g, J)$ has negative $\lambda(m)$-pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants $\lambda(m)$ satisfy $\lambda(6) = 0.9330...$, $\lim_{m \to +\infty} \lambda(m) = \tfrac{11}{12} = 0.9166...$, and $m \mapsto \lambda(m)$ is decreasing. This extends to the even-dimensional case the results of Brin-Gromov who proved ergodicity of the unitary frame flow on negatively-curved compact K\"ahler manifolds of odd complex dimension.