The transverse Chern-Ricci flow

TL;DR

This paper introduces a transverse version of the Chern-Ricci flow for transversely Hermitian foliations, proving long-term existence and convergence under certain conditions, and extending previous results to a foliated setting.

Contribution

It develops the transverse Chern-Ricci flow for foliations, establishing existence, convergence, and maximal existence time results, extending classical complex geometric flows to foliated manifolds.

Findings

Flow exists for all time when basic first Bott-Chern class is zero.

Flow converges smoothly to a transverse Hermitian metric with zero transverse Chern-Ricci form.

Determines maximal existence time in the general case.

Abstract

We introduce transverse Chern-Ricci flow for transversely Hermitian foliations, which is analogous to the Chern-Ricci flow. We show that when $\mathcal{F}$ is homologically orientable and the basic first Bott-Chern class is zero, starting at any transversely Hermitian metric the flow exists for all time and as $t\rightarrow \infty$ converges smoothly to a transversely Hermitian metric $\omega_\infty$ with the transverse Chern-Ricci form $\rho^T(\omega_\infty)=0$. We also determine the maximal existence time of the flow in the general case. These are foliated version of results of Gill and Tosatti-Weinkove, and also extend recent work of Bedulli-He-Vezzoni.