Hermitian Curvature flow on unimodular Lie groups and static invariant metrics

TL;DR

This paper studies the Hermitian curvature flow on complex unimodular Lie groups, showing long-term existence, convergence to solitons, and characterizing static metrics, including on semisimple and nilpotent groups, with implications for non-Kähler manifolds.

Contribution

It provides a comprehensive analysis of HCF on unimodular Lie groups, establishing existence, convergence, and uniqueness of static solutions, and extends results to abelian complex structures.

Findings

Flow always exists for all positive times.

Scaled metrics converge to non-flat solitons.

Unique static solutions exist on semisimple groups.

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 $\partial_tg_{t}=-{\rm Ric}^{1,1} (g_t)$. 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-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.