On the Chern‐Ricci flow and its solitons for Lie groups

Abstract

Abstract. 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.

Laplacian flow of homogeneous G2-structures and its solitons

“…It follows from the above theorem that any simply connected algebraic soliton is indeed a Laplacian soliton. The concept of algebraic soliton has been very fruitful in the study of homogeneous Ricci solitons since its introduction in , we refer to [, Sections 5.2 and 5.4] for a quick overview (see also for other curvature flows). Nothing changes by allowing a derivation of the form in the definition of algebraic soliton since must necessarily hold (see [, Remark 7]).…”

“…The concept of algebraic soliton has a long and fruitful history in the Ricci flow case, due perhaps to its neat definition as a combination of geometric and algebraic aspects of . It has also been a useful tool to address the existence problem of soliton structures for general curvature flows in almost‐Hermitian geometry (see ), the symplectic curvature flow (see ) and the Chern–Ricci flow (see ). As in any of these cases, a natural question is how special are algebraic solitons among homogeneous Laplacian solitons.…”

“…The Laplacian flow solution ϕ(t) and the bracket flow solution μ(t) have the same maximal interval of time existence, say (After some preliminaries on G 2 geometry in Section 2, we adapt in Section 3 the machinery developed in [24] to the Laplacian flow case; the results so obtained include (see Sections 3.3 and 3.4 for more precise statements).(i) The norm |Δ ϕ(t) ϕ(t)| ϕ(t) of the velocity of the flow must blow up at any finite-time singularity (compare with [28, Theorem 1.6]).(ii) If μ(t) converges to a Lie bracket λ, as t → T ± , and there is a positive lower bound for the (Lie) injectivity radii of the G-invariant metrics g ϕ(t) on G/K, then (G λ /K λ , ϕ) is a Laplacian soliton (possibly non-homeomorphic to G/K) and (G/K, ϕ(t)) converges in the pointed (or Cheeger-Gromov) sense to (G λ /K λ , ϕ), as t → T ± .(iii) The following conditions on a simply connected (G/K, ϕ) are equivalent.(a) The operator Q ϕ such that θ(Q ϕ )ϕ = Δ ϕ ϕ satisfiesthat is, (G/K, ϕ) is an algebraic soliton.where X D denotes the vector field on G/K defined by the one-parameter subgroup of automorphisms of G attached to the derivation D (in particular, (G/K, ϕ) is a Laplacian soliton).The concept of algebraic soliton has a long and fruitful history in the Ricci flow case, due perhaps to its neat definition as a combination of geometric and algebraic aspects of (G/K, ϕ). It has also been a useful tool to address the existence problem of soliton structures for general curvature flows in almost-Hermitian geometry (see [23]), the symplectic curvature flow (see [9,26]) and the Chern-Ricci flow (see [25]). As in any of these cases, a natural question is how special are algebraic solitons among homogeneous Laplacian solitons.…”

Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures

“…Let us also define d n 0,0 = 1, d n 1,0 = d n 0,1 = 0, so that D n 0 = 1, D n 1 = 0. Using this together with (23), we readily obtain that (24) β n k = D n k−2 + 2D n k−1 + D n k , k ≥ 2.…”

“…Left invariant LCK or LCS structures on nilpotent Lie groups have been thoroughly studied in [6,39], therefore we will focus on the class of almost abelian Lie groups. This class has appeared recently in several different contexts (see for instance [4,7,9,24,25]). LCK and LCS structures on Lie groups and Lie algebras (and also in their compact quotients by discrete subgroups, if they exist) have been considered by several authors lately.…”