Hermitian Curvature flow on unimodular Lie groups and static invariant metrics

DOI10.1090/TRAN/8068zbMATH Open1509.53107arXiv1807.00059WikidataQ115280241 ScholiaQ115280241MaRDI QIDQ6303703

Mattia Pujia, Ramiro Lafuente, Luigi Vezzoni

Publication date: 29 June 2018

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 $partia{l}_t{g}_t=-{mRic}^{1,1}(g_t)$. The solution $g_t$ always exist for all positive times, and $(1+t{)}^{-1}{g}_t$ converges as $toinfty$ in Cheeger-Gromov sense to a non-flat left-invariant soliton $$. 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-K"ahler 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.

Mathematics Subject Classification ID

Differential geometry of homogeneous manifolds (53C30) General geometric structures on manifolds (almost complex, almost product structures, etc.) (53C15) Other connections (53B15) Flows related to complex manifolds (e.g., Kähler-Ricci flows, Chern-Ricci flows) (53E30)

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