The definition of a simply-connected Riemann surface is the hyperbolic type if there exists a non-constant and bounded subharmonic function; the compact Riemann surface is called the ecliptic type. And the else is the parabolic type. I try to prove the
$\mathbb{C}^{*}$ is a hyperbolic type.
I have proved that
$\mathbb{C}$ is a parabolic type.
I want to use this conclusion, but I do not sure my method without error.
If $\mathbb{C}^{*}$ is not an ecliptic type, it is a hyperbolic type. Then there exists a non-constant and bounded subharmonic function $u$ such that $u: \mathbb{C}^{*} \rightarrow \mathbb{R}$. And I consider map $e^x: \mathbb{C} \rightarrow \mathbb{C}^{*}$, then the left$\psi=u\circ e^x$, $$\psi: \mathbb{C}\rightarrow \mathbb{R}$$ is a non-constant and bounded subharmonic function, contradiction.