Hermitian curvature flow on unimodular Lie groups and static invariant metrics

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 Ricciflow type equation ∂tgt = −Ric 1,1 (gt). The solution gt always exist for all positive times, and (1 + t) −1 gt converges as t → ∞ in Cheeger-Gromov sense to a non-flat left-invariant soliton (Ḡ,ḡ). 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 in [16] for the pluriclosed flow. In the last part of the paper we study HCF on Lie groups with abelian complex structures.

Positive Hermitian Curvature Flow on special linear groups and perfect solitons

“…Thus, v(y, z) ∈ int T (y,z) D, the interior of the tangent cone of D at (y, z), for all (y, z) ∈ ∂D. It follows that D is an invariant subset for the ODE (12). Now let (y, z) be a solution to (12) in D, which exists for all positive times as it remains in the compact set D. In this case, ż ≤ 0 if and only if y ≤ 1 n+1 (nz 2 + 1).…”

“…It follows that D is an invariant subset for the ODE (12). Now let (y, z) be a solution to (12) in D, which exists for all positive times as it remains in the compact set D. In this case, ż ≤ 0 if and only if y ≤ 1 n+1 (nz 2 + 1). Now if (y, z) is a solution to ( 12) with (y 0 , z 0 ) ∈ D, then it exists for all positive times as it remains in the compact subset D. Thus, since ẏ ≤ 0, the Poincaré-Bendixson Theorem implies that the omega limit must be {(0, 0)}, consisting of the unique fixed point inside D.…”

“…Theorem 1.5 therefore demonstrates that the theory of the HCF + differs substantially to that of others in the HCF family. In [12], Lafuente, Pujia, and Vezzoni studied the first version of the HCF introduced by Streets and Tian in [23] as the gradient flow of a particular functional. They showed (in contrast to Theorem 1.5) that on complex unimodular Lie groups, left-invariant semi-algebraic solitons are algebraic and are unique up to homothety.…”

“…Several members of the HCF family are actively being studied in the homogeneous case. We refer the reader to [1,4,7,8,12,18,19,20,21,22,24].…”

Hermitian Curvature Flow on Compact Homogeneous Spaces

“…Given a connected complex Lie group G acting transitively, effectively and holomorphically on a complex manifold M , a simple observation shows that M does not carry any G-invariant Hermitian metric unless the isotropy is finite. More recently in [9] Lafuente, Pujia and Vezzoni considered the behaviour of a general HCF in the space of left-invariant Hermitian metrics on a complex unimodular Lie group.…”