Paper · Oct 9, 2025

Positive Hermitian curvature flow on 2-step nilpotent Lie groups

Ettore Lo Giudice

math.DGarXiv:2510.08846
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Abstract

We study the positive Hermitian curvature flow for left-invariant metrics on -step nilpotent Lie groups with a left-invariant complex structure . We describe the long-time behavior of the flow under the assumption that is contained in the center of . We show that under our assumption the flow exists for all positive and converges, in the Cheeger-Gromov topology, to a -step nilpotent Lie group with a non flat semi-algebraic soliton. Moreover, we prove that, in our class of Lie groups, there exists at most one semi-algebraic soliton solution, up to homothety. Similar results were proved by M. Pujia and J. Stanfield for nilpotent complex Lie groups. In the last part of the paper we study the Hermitian curvature flow for the same class of Lie groups.

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Submitted
Oct 9, 2025
Subjects
math.DG
Generalized & G₂ flowsHomogeneous spacesLong-time existence & convergence

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    Topics Homogeneous spaces · Shared terms “nilpotent Lie”, “step nilpotent”, “Lie groups”

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.