Conformal extremal metrics and constant scalar curvature

Abstract

Let $M$ be a compact complex manifold of dimension $n\ge 2$. We prove that for any Hermitian metric $\omega$ on $M$, there exists a unique smooth function $f$ (up to additive constants) such that the conformal metric ${\omega }_g=e^f\omega$ solves the fourth-order nonlinear PDE $${\square }_g^{\ast }(s_g|s_g{|}^{n-2})=0,$$ where $s_g$ is the Chern scalar curvature of ${\omega }_g$, and ${\square }_g^{\ast }$ denotes the formal adjoint of the complex Laplacian ${\square }_g={\mathrm{tr}}_{{\omega }_g}\sqrt{-1}\partial \bar{\partial }$ with respect to ${\omega }_g$. This equation arises as the Euler-Lagrange equation of the $n$-Calabi functional $$C_n({\omega }_g)=\int |s_g{|}^n\frac{{\omega }_g^n}{n!}$$ within the conformal class of ${\omega }_g$. Moreover, we show that the critical metric ${\omega }_g$ minimizes the $n$-Calabi functional within the conformal class $[\omega ]$. In particular, if ${\omega }_g$ is a Gauduchon metric, then ${\omega }_g$ has constant Chern scalar curvature.