Hermitian curvature flow on unimodular Lie groups and static invariant metrics

Abstract:

We investigate the Hermitian curvature flow (HCF) of leftinvariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation $\partial _tg_{t}=-{\mathrm { Ric}}^{1,1}(g_t)$. The solution $g_t$ always exists for all positive times, and $(1 + t)^{-1}g_t$ converges as $t\to \infty$ in Cheeger–Gromov sense to a nonflat 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 of Enrietti, Fino, and the third author [J. Symplectic Geom. 10 (2012), no. 2, 203–223] for the pluriclosed flow. In the last part of the paper we study the HCF on Lie groups with abelian complex structures.