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 \(\begin{aligned} \square _g^*(s_g|s_g|^{n-2})=0, \end{aligned}\) where \(s_g\) is the Chern scalar curvature of \(\omega _g\) , and \({\square }_g^*\) denotes the formal adjoint of the complex Laplacian \({\square }_g=\textrm{tr}_{\omega _g}{\sqrt{-1}}\partial {\overline{\partial }}\) with respect to \(\omega _g\) . This equation arises as the Euler-Lagrange equation of the n-Calabi functional \(\begin{aligned} C_{n}(\omega _g)=\int |s_g|^n\frac{\omega _g^n}{n!} \end{aligned}\) 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.
