This paper is concerned with Chern-Ricci flow evolution of left-invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger-Gromov) sense to a Chern-Ricci soliton. We give some results on the Chern-Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgroup. We have also found a Chern-Ricci soliton hermitian metric on most of the complex surfaces which are solvmanifolds, including an unexpected shrinking soliton example. Contents 1. Introduction 1 2. Chern-Ricci form 3 3. Chern-Ricci flow 4 Bracket flow 5 4. Chern-Ricci solitons 6 5. Convergence 7 6. Almost-abelian Lie groups 10 7. Lie groups of dimension 4 13 References 16This research was partially supported by grants from CONICET, FONCYT and SeCyT (Univ. Nac. Córdoba).1
Chern-Ricci formLet (M, J, ω, g) be a 2n-dimensional hermitian manifold, where ω = g(J•, •). The Chern connection is the unique connection ∇ on M which is hermitian (i.e. ∇J = 0, ∇g = 0) and its torsion satisfies T 1,1 = 0. In terms of the Levi Civita connection D of g, the Chern connection is given by g(∇ X Y, Z) = g(D X Y, Z) − 1 2 dω(JX, Y, Z). We refer to e.g. [V2, (2.1)], [DV, (2.1)] and [TW1, Section 2] for different equivalent descriptions. Note that ∇ = D if and only if (M, J, ω, g) is Kähler.The Chern-Ricci form p = p(J, ω, g) is defined by
