The transverse Chern-Ricci flow

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 $ω_\infty$ with the transverse Chern-Ricci form $ρ^T(ω_\infty)=0$. 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.

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