On the ergodicity of unitary frame flows on Kähler manifolds

Abstract

AbstractLet$(M,g,J)$be a closed Kähler manifold with negative sectional curvature and complex dimension$m := \dim _{\mathbb {C}} M \geq 2$. In this article, we study theunitary 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.