This paper considers maps from pseudo-Hermitian manifolds to Kähler manifolds and introduces partial energy functionals for these maps. First, the authors obtain a foliated Lichnerowicz type result on general pseudo-Hermitian manifolds, which generalizes a related result on Sasakian manifolds by Shen–Shen–Zhang (2013). Next, the authors investigate critical maps of the partial energy functionals, which are referred to as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\overline \partial}_{b}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mover> <mi mathvariant="normal">∂</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow> <mi>b</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>-harmonic maps and ∂b-harmonic maps. The authors give a foliated result for both <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\overline \partial}_{b}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mover> <mi mathvariant="normal">∂</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow> <mi>b</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>- and ∂b-harmonic maps, generalizing a foliated result of Petit (2002) for harmonic maps. Then the authors are able to generalize Siu’s holomorphicity result for harmonic maps by Siu (1980) to the case for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\overline \partial}_{b}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mover> <mi mathvariant="normal">∂</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow> <mi>b</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>- and ∂b-harmonic maps.

On ${\overline \partial}_{b}$-Harmonic Maps from Pseudo-Hermitian Manifolds to Kähler Manifolds

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发表日期：

2026年1月7日

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Yuxin Dong,
Hui Liu,
Biqiang Zhao

摘要

This paper considers maps from pseudo-Hermitian manifolds to Kähler manifolds and introduces partial energy functionals for these maps. First, the authors obtain a foliated Lichnerowicz type result on general pseudo-Hermitian manifolds, which generalizes a related result on Sasakian manifolds by Shen–Shen–Zhang (2013). Next, the authors investigate critical maps of the partial energy functionals, which are referred to as ${\overline \partial}_{b}$ ${\bar{\partial}}_{b}$ -harmonic maps and ∂_b-harmonic maps. The authors give a foliated result for both ${\overline \partial}_{b}$ ${\bar{\partial}}_{b}$ - and ∂_b-harmonic maps, generalizing a foliated result of Petit (2002) for harmonic maps. Then the authors are able to generalize Siu’s holomorphicity result for harmonic maps by Siu (1980) to the case for ${\overline \partial}_{b}$ ${\bar{\partial}}_{b}$ - and ∂_b-harmonic maps.

On \({\overline \partial}_{b}\)-Harmonic Maps from Pseudo-Hermitian Manifolds to Kähler Manifolds

摘要