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

TL;DR

This paper proves ergodicity and mixing of the unitary frame flow on certain negatively curved K"ahler manifolds with even complex dimension, extending previous results from odd dimensions under specific curvature pinching conditions.

Contribution

It establishes ergodicity of the unitary frame flow for even-dimensional K"ahler manifolds with negative curvature under explicit pinching conditions, extending prior odd-dimensional results.

Findings

Ergodicity and mixing of the unitary frame flow under curvature pinching.

Explicit bounds for the pinching constant (m) for different dimensions.

Extension of previous odd-dimensional results to even-dimensional cases.

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.