A question about complex Laplacian on compact Hermitian manifolds

Question

Let $(M,g)$ be a complex $n$-dim compact connected Hermitian manifold, and $\Delta_c(\cdot):=g^{i\bar{j}}\partial_i\partial_{\bar{j}}(\cdot)$ the complex Laplacian acting on smooth functions on $M$. It is well-known that this $\Delta_c$ is in general not equal to the usual Laplacian and this holds exactly when the metric $g$ is balanced. (due to Gauduchon?)

My question is, does this complex Laplacian $\Delta_c$ behave like the usual Laplacian? To be more precise, I have two related questions.

1. Does $\int_M\Delta_c(f)=0?$ for any $f$?

2. If so, given $f$, does the function $\Delta_c(u)=f$ have a solution $u$ if and only if $\int_M f=0$?

Many thanks in advance!

Answer 1

For your first question, note that (let $\omega$ be the Hermitian form)

$$\int_M\Delta_c(f) \omega^n=\int_M(\mathrm{tr}_\omega \sqrt{-1}\partial \overline{\partial}f)\omega^n=n\int_M \sqrt{-1}\partial \overline{\partial}f \wedge \omega^{n-1}$$

Using Stoke's formula, we deduce that

$$\int_M \partial \overline{\partial}f \wedge \omega^{n-1}=\int_M \overline{\partial}f \wedge \mathrm{d}\omega^{n-1}=\int_M \overline{\partial}f \wedge \partial \omega^{n-1}=\int_M f \wedge \partial \overline{\partial} \omega^{n-1}$$

Then $$ \int_M\Delta_c(f) \omega^n=n \int_M f \wedge \sqrt{-1}\partial \overline{\partial} \omega^{n-1} $$

And $\int_M\Delta_c(f) \omega^n=0$ for all $f$ iff $\partial \overline{\partial} \omega^{n-1}=0$. Such metric is called a $\textbf{Gauduchon metric}$.

The answer to your second question is positive and it was first proved by Gauduchon in "Le théorème de l'excentricité nulle. C. R. Acad. Sci. Paris 285, 387–390 (1977)".

A more accessible reference is "The Monge–Ampère equation for non-integrable almost complex structures" by Chu, Tosatti and Weinkove. They proved the results in Theorem 2.1, Theorem 2.2 in their paper (for almost Hermitian manifolds):

Answer 2

You can search the topics related to Chern-Laplacian on $(M,\omega)$ $$ \Delta^{C h} f=\Delta_d f+(d f, \theta)_\omega. $$ Actually it's just a Hodge- Laplacian plus a linear gradient term, here $\theta$ is the Lee form of $\omega$. For the first question, this requires the metric to be Gauduchon ($d^* \theta=0$), so we can integrate the gradient term by part to get that $$ \int_M \Delta^{Ch}f=0. $$ For the second question, for $$ \Delta^{Ch}u=f, $$ you can always find a solution when the metric is in the conformal class of the original metric, because in the conformal class there must be a Gauduchon metric, so if we consider $$ \left(\Delta^{C h}\right)^* g=\Delta_d g-(d g, \theta)_\eta $$ here $\eta$ is the new Gauduchon metric and $\theta$ is the Lee form of $\eta$. We will have $$ 0=\int_M u\left(\Delta^{C h}\right)^* u =\int_M\left(|\nabla u|^2-\frac{1}{2}\left(d u^2, \theta\right)\right) =\int_M|\nabla u|^2 , $$ which means kernel of $\left(\Delta^{C h}\right)^*$ is the constant set. Since the integral of $f$ is zero, it means that $f \in\left(\operatorname{ker}\left(\Delta^{C h}\right)^*\right)^{\perp}=\operatorname{imm} \Delta^{C h}$, which proves the existence of the solution.

And when the manifold is kahler, the gradient term will vanish, then the Chern connection is the same as Levi-Civita connection, you can refer to Daniele Angella's work on this topic, for example On Gauduchon connections with Kähler-like curvature

Besides, the prescribed Chern scalar curvature problem is related to Kazdan-Warner equation, but there are some open problems in this topic, you can read the following paper if you are interested On Chern-Yamabe problem

The prescribed Chern scalar curvature problem

The prescribed Gauduchon scalar curvature problem in almost Hermitian geometry