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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.15415 |
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| _version_ | 1866913850759053312 |
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| author | Yang, Xiaokui Zhang, Kaijie |
| author_facet | Yang, Xiaokui Zhang, Kaijie |
| contents | Let $M$ be a compact complex manifold of dimension $n\geq 2$. We prove that for any Hermitian metric $ω$ on $M$, there exists a unique smooth function $f$ (up to additive constants) such that the conformal metric $ω_g =e^f ω$ solves the fourth-order nonlinear PDE
$$\square_g^*(s_g|s_g|^{n-2})=0,$$
where $s_g$ is the Chern scalar curvature of $ω_g$, and $\square_g^*$ denotes the formal adjoint of the complex Laplacian $\square_g=\mathrm{tr}_{ω_g}\sqrt{-1}\partial\bar\partial$ with respect to $ω_g$. This equation arises as the Euler-Lagrange equation of the $n$-Calabi functional
$$C_{n}(ω_g)=\int |s_g|^n\frac{ω_g^n}{n!}$$ within the conformal class of $ω_g$. Moreover, we show that the critical metric $ω_g$ minimizes the $n$-Calabi functional within the conformal class $[ω]$. In particular,
if $ω_g$ is a Gauduchon metric, then $ω_g$ has constant Chern scalar curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15415 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conformal extremal metrics and constant scalar curvature Yang, Xiaokui Zhang, Kaijie Differential Geometry 53C55 Let $M$ be a compact complex manifold of dimension $n\geq 2$. We prove that for any Hermitian metric $ω$ on $M$, there exists a unique smooth function $f$ (up to additive constants) such that the conformal metric $ω_g =e^f ω$ solves the fourth-order nonlinear PDE $$\square_g^*(s_g|s_g|^{n-2})=0,$$ where $s_g$ is the Chern scalar curvature of $ω_g$, and $\square_g^*$ denotes the formal adjoint of the complex Laplacian $\square_g=\mathrm{tr}_{ω_g}\sqrt{-1}\partial\bar\partial$ with respect to $ω_g$. This equation arises as the Euler-Lagrange equation of the $n$-Calabi functional $$C_{n}(ω_g)=\int |s_g|^n\frac{ω_g^n}{n!}$$ within the conformal class of $ω_g$. Moreover, we show that the critical metric $ω_g$ minimizes the $n$-Calabi functional within the conformal class $[ω]$. In particular, if $ω_g$ is a Gauduchon metric, then $ω_g$ has constant Chern scalar curvature. |
| title | Conformal extremal metrics and constant scalar curvature |
| topic | Differential Geometry 53C55 |
| url | https://arxiv.org/abs/2505.15415 |