is the twice punctured complex plane parabolic or hyperbolic? In this sense: does $\mathbb{C}-\{0,1\}$ admit a nonconstant, negative, subharmonic function?
Thanks,
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
Sign up to join this communityA subharmonic function which is bounded from above near a polar set automatically extends through it. Therefore, any negative subharmonic function on $\mathbb C \setminus\{0,1\}$ induces a negative subharmonic function on $\mathbb C$, and by Liouville theorem, it must be constant.
So the answer to your question is no.
The $j$-invariant in the theory of elliptic curves is a covering map from the upper half plane to the complex plane punctured at $0$ and $1$. See Koblitz, Introduction to Elliptic Curves and Modular Forms.