On global rigidity of transversely holomorphic Anosov flows

TL;DR

This paper investigates the structure of transversely holomorphic Anosov flows on compact manifolds, revealing conditions under which they are equivalent to well-known geometric flows like torus automorphisms or hyperbolic geodesic flows.

Contribution

It establishes a classification of topologically transitive transversely holomorphic Anosov flows in five dimensions, linking them to classical geometric models.

Findings

Unstable and stable distributions are integrable to complex manifolds with holomorphic flow action.

In one complex dimension, distributions are uniquely integrable to affine complex lines.

Topologically transitive flows are orbit equivalent to either torus automorphisms or hyperbolic geodesic flows.

Abstract

In this paper, we study transversely holomorphic flows, i.e. those whose holonomy pseudo-group is given by biholomorphic maps. We prove that for Anosov flows on smooth compact manifolds, the strong unstable (respectively, stable) distribution is integrable to complex manifolds, on which the flow acts holomorphically. Furthermore, assuming its complex dimension to be one, it is uniquely integrable to complex affine one-dimensional manifolds, each moreover affinely diffeomorphic to $\mathbb C$, on which the flow acts affinely. In this case, the weak stable (respectively, unstable) foliation is transversely holomorphic, and even transversely projective if the flow is assumed to be topologically transitive. By combining these facts in low dimensions, our main result is as follows : if a transversely holomorphic Anosov flow on a smooth compact five-dimensional manifold is topologically transitive, then it is either $C^\infty$-orbit equivalent to the suspension of a hyperbolic automorphism of a complex torus, or, up to finite covers, $C^\infty$-orbit equivalent to the geodesic flow of a compact hyperbolic manifold.