Conformal extremal metrics and constant scalar curvature

Abstract

Let $M$ be a compact complex manifold of dimension $n\geq 2$. 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 $$\square_g^*(s_g|s_g|^{n-2})=0,$$ where $s_g$ is the Chern scalar curvature of $\omega_g$, and $\square_g^*$ 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.