Positive Hermitian curvature flow on complex 2-step nilpotent Lie groups
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Mattia Pujia
Positive Hermitian curvature flow on complex 2-step nilpotent Lie groups Received: 22 February 2020 / Accepted: 11 August 2020 Abstract. We study the positive Hermitian curvature flow of left-invariant metrics on complex 2-step nilpotent Lie groups. In this setting we completely characterize the long-time behaviour of the flow, showing that normalized solutions to the flow subconverge to a nonflat algebraic soliton, in Cheeger–Gromov topology. We also exhibit a uniqueness result for algebraic solitons on such Lie groups.
Introduction In 2011, Streets and Tian introduced a new family of parabolic flows generalizing the Kähler-Ricci flow to the Hermitian setting [26]. More precisely, given a complex manifold X , any flow in the family evolves a Hermitian metric g on X via ∂t gt = −S(gt ) + Q(gt ), gt |0 = g,
(1)
where S(g) is the second Chern–Ricci curvature tensor of g on X and Q(g) is a (1, 1)-symmetric tensor quadratic in the torsion of the Chern connection. The flows belonging to such a family are usually called Hermitian curvature flows (HCFs for short). In [26], Streets and Tian chose the tensor Q in order to obtain a gradient flow, stable near Kähler-Einstein metrics with non-positive scalar curvature and satisfying many other analytical properties; while, in the subsequent paper [25], ¯ = 0) (see Q was chosen so that the pluriclosed condition was preserved (∂ ∂ω also [23,24,27,28]). On the other hand, since the tensor Q does not affect the parabolicity of the evolution Eq. (1), different choices of Q can be performed in order to preserve other properties. In [30], following Streets and Tian’s approach, Ustinovskiy introduced a HCF preserving both the Griffiths-positivity and the dual Nakano-positivity of the tangent bundle. More precisely, given a compact Hermitian manifold (X, g), Ustinovskiy considered the evolution equation t ), ∂t gt = −S(gt ) − Q(g
gt |0 = g.
(2)
M. Pujia (B): Dipartimento di Matematica G. Peano, Università di Torino, Via Carlo Alberto 10, 10123 Turin, Italy e-mail: [email protected]; [email protected] Mathematics Subject Classification: Primary 53C44 · Secondary 53C15 · 53C07 · 53B15
https://doi.org/10.1007/s00229-020-01251-w
M. Pujia
Here, we denote by: ∇ the Chern connection of (X, g), the curvature tensor of ∇, S(g) the (1, 1)-symmetric tensor ¯
Si j¯ = g kl kli¯ j¯, and Q(g) the tensor given by ¯ = g kl¯g m n¯ T ¯ T¯ , 2Q ij km j l ni ¯ l and T l are the components of the torsion of ∇. We will where Tkm j¯ := gl j¯ Tkm km refer to (2) as to the positive Hermitian curvature flow (HCF+ for short) and we set
K (g) := S(g) + Q(g), for any Hermitian metric g on X . The aim of the present paper is to study the behaviour of the HCF+ on complex nilpotent Lie groups when the initial metrics are left-invariant. Although in the non-compact case existence and uniqueness of left-invariant solutions to (2) are not always guaranteed, in our setting the invariance by biholomorphisms of t
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