Conformal extremal metrics and constant scalar curvature
Xiaokui Yang, Kaijie Zhang
TL;DR
The paper extends Calabi-type energy methods to Hermitian (non-Kähler) manifolds by introducing the $p$-Calabi functional and studying $p$-conformal extremal metrics. It derives the Euler–Lagrange equations $\square_g^*(s_g|s_g|^{p-2})=\frac{n-p}{np}\left(|s_g|^p-\int|s_g|^p\frac{\omega_g^n}{n!}\right)$ and proves that, in particular, the homogeneous case $p=n$ has a unique solution in each conformal class, which minimizes $C_n$; if the extremal metric is Gauduchon, it has constant Chern scalar curvature. The authors show that the $n$-conformal extremal metric can be written as $\omega_E=e^f\omega_G$ with $f$ solving a linear Poisson-type equation on the Gauduchon representative, yielding existence, uniqueness (up to constants), and a precise energy-minimization statement. They also prove rigidity and structural results: constant nonzero $s_p$ implies Gauduchon, and Gauduchon metrics with constant $s_g$ are $p$-conformal extremal for all $p>1$, linking non-Kähler scalar curvature geometry to conformal variational problems.
Abstract
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.