On the ergodicity of unitary frame flows on Kähler manifolds

Abstract

Let $(M,g,J)$ be a closed Kähler manifold with negative sectional curvature and complex dimension $m:={\dim }_{\mathbb{C}}M\ge 2$. 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 $m\ge 6$ is even, and $m\ne 28$, 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 $\lambda (6)=\mathrm{0.9330...}$, ${\lim }_{m\to +\infty }\lambda (m)=\frac{11}{12}=\mathrm{0.9166...}$, 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 Kähler manifolds of odd complex dimension.

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Cited by in corpus (3)

• On the ergodicity of unitary frame flows on Kähler manifolds
• On the ergodicity of the frame flow on even-dimensional manifolds
• Correspondence between Pestov and Weitzenböck identities