We study the positive Hermitian curvature flow for left-invariant metrics on 2-step nilpotent Lie groups G with a left-invariant complex structure J. We describe the long-time behavior of the flow under the assumption that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$J[\mathfrak {g}, \mathfrak {g}]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <mo stretchy="false">[</mo> <mi mathvariant="fraktur">g</mi> <mo>,</mo> <mi mathvariant="fraktur">g</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> is contained in the center of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathfrak {g}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">g</mi> </math></EquationSource> </InlineEquation>. We show that under our assumption the flow <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$g_{t}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>g</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> exists for all positive t and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(G,(1+t)^{-1}g_{t})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>g</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> converges, in the Cheeger-Gromov topology, to a 2-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 [<CitationRef CitationID="CR21">21</CitationRef>, <CitationRef CitationID="CR24">24</CitationRef>]. In the last part of the paper we study the Hermitian curvature flow for the same class of Lie groups.

Positive Hermitian curvature flow on 2-step nilpotent Lie groups

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2026年3月9日

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Ettore Lo Giudice

摘要

We study the positive Hermitian curvature flow for left-invariant metrics on 2-step nilpotent Lie groups G with a left-invariant complex structure J. We describe the long-time behavior of the flow under the assumption that $J[\mathfrak {g}, \mathfrak {g}]$ $J [g, g]$ is contained in the center of $\mathfrak {g}$ $g$ . We show that under our assumption the flow $g_{t}$ $g_{t}$ exists for all positive t and $(G,(1+t)^{-1}g_{t})$ $(G, {(1 + t)}^{- 1} g_{t})$ converges, in the Cheeger-Gromov topology, to a 2-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 [21, 24]. In the last part of the paper we study the Hermitian curvature flow for the same class of Lie groups.