If $f : A \subset M \to \mathbb{R}$ is a $C^{\infty}(M)$ function then $dd^c f = 0 \Rightarrow f = cte.$

Question

I am trying to prove the following:

If $f : A \subset M \to \mathbb{R}$ is a $C^{\infty}(M)$ function such that $dd^c f = 0$ then $f = c,$ where $M$ is a compact Kähler manifold, $c\in \mathbb{R}$. We remember that $dd^c = 2i\partial\overline{\partial}$ and $\partial$, $\overline{\partial}$ are de Dolbeault operators.

Answer 1

Hints:

1. Every solution to $dd^c f=0$ is harmonic.
2. Every harmonic function on a compact manifold is constant.