Conformal metrics

Question

While studying complex structures on Riemannian manifolds I found some trouble in proving this claim:

"There is no metric conformal to the Euclidean one on $\mathbb{C}-{0}$ whose Gaussian curvature is upper bounded by a negative constant".

There's a hint also, about using the fact that there are non-constant holomorphic functions on $\mathbb{C}$ whose codomain is $\mathbb{C}-{0}$, like $f(z)=e^{z}$, but I really don't know how to use it properly. I think it may be linked to Picard's theorem, but I'm not sure about that. Can anyone help me to figure it out? Thank you.

Answer 1

I don't know how much complex analysis you know, but here is an almost one-liner proof:

If you have such a metric, you also get one on $\mathbb{C}$ by pulling back via $\exp\colon\mathbb{C}\to\mathbb{C}-\{0\}$. Hence every self-map of $\mathbb{C}$ has the distance-decreasing property (Ahlfors-Schwarz-Pick) which is obviously a contradiction.