Conformal extremal metrics and constant scalar curvature

2025/05/21 by Xiaokui Yang, Kai Zhang, Yang, Xiaokui +1
Physics and Astronomy · Mathematics · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2505.15415

Abstract

Let M be a compact complex manifold of dimension n≥ 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=tr_ωg√(-1)∂∂ with respect to ω_g. This equation arises as the Euler-Lagrange equation of the n-Calabi functional C_n(ω_g)=∫ |s_g|ⁿ(ω_gⁿ)/(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.

Citations

• On the Gauduchon Curvature of Hermitian Manifolds
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• Scalar curvatures in almost Hermitian geometry and some applications
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• Conformal metrics of constant scalar curvature with unbounded volumes