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 ∂ t g t = − R i c 1 , 1 ( g t ) \partial _tg_{t}=-{\mathrm { Ric}}^{1,1}(g_t) . The solution g t g_t always exists for all positive times, and ( 1 + t ) − 1 g t (1 + t)^{-1}g_t converges as t → ∞ t\to \infty in Cheeger–Gromov sense to a nonflat left-invariant soliton ( G ¯ , g ¯ ) (\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.