Hermitian Curvature flow on unimodular Lie groups and static invariant metrics

Abstract

We investigate the Hermitian curvature flow (HCF) of left-invariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation ${\partial }_t{g}_t=-{\mathrm{Ric}}^{1,1}(g_t)$. The solution $g_t$ always exist for all positive times, and $(1+t{)}^{-1}{g}_t$ converges as $t\to \infty$ in Cheeger-Gromov sense to a non-flat left-invariant soliton $(\bar{G},\bar{g})$. Moreover, up to homotheties on each of these groups there exists at most one left-invariant soliton solution, which is a static Hermitian metric if and only if the group is semisimple. In particular, compact quotients of complex semisimple Lie groups yield examples of compact non-Kähler manifolds with static Hermitian metrics. We also investigate the existence of static metrics on nilpotent Lie groups and we generalize a result in \cite{EFV} for the pluriclosed flow. In the last part of the paper we study HCF on Lie groups with abelian complex structures.

25 pages. Revised version. To appear in TAMS