On the pluriclosed flow on Oeljeklaus-Toma manifolds

TL;DR

This paper studies the behavior of the pluriclosed flow on Oeljeklaus-Toma manifolds, classifying invariant metrics and analyzing long-term geometric convergence to algebraic solitons.

Contribution

It classifies invariant pluriclosed metrics that lift to algebraic solitons and describes the flow's long-term convergence to a torus and algebraic solitons.

Findings

Flow solutions collapse to a torus in Gromov-Hausdorff sense.

Lifted metrics converge to algebraic solitons in Cheeger-Gromov sense.

Classification of metrics lifting to algebraic solitons.

Abstract

We investigate the pluriclosed flow on Oeljeklaus-Toma manifolds. We parametrize left-invariant pluriclosed metrics on Oeljeklaus-Toma manifolds and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution $\omega_t$ which once normalized collapses to a torus in the Gromov-Hausdorff sense. Moreover the lift of $\tfrac{1}{1+t}\omega_t$ to the universal covering of the manifold converges in the Cheeger-Gromov sense to $(\mathbb H^r\times\mathbb C^s, \tilde{\omega}_{\infty})$ where $\tilde{\omega}_{\infty}$ is an algebraic soliton.