How do we prove that any Riemann surface which is diffeomorphic to a punctured disc can be holomorphically embedded in $\mathbb{C}$?
The reason I am thinking about this is because I was trying to classify complex structures on a punctured disc. Any hints will be appreciated. Thanks.
Such a Riemann surface $X$ has fundamental group $\Bbb Z$. By the uniformisation theorem, its universal cover $U$ is either $\Bbb C$ or the upper half plane, and $X$ is the quotient of $U$ by an automorphism $\gamma$ of infinite order. Up to inner automorphisms, if $U=\Bbb C$ then $\gamma$ is conjugate to $z\mapsto z+1$, and then $X$ is conformally equivalent to $\Bbb C^*$. If $U$ is the upper half plane, then we can take $\gamma:z\mapsto z+1$ or $\gamma: z\mapsto \lambda z$ ($\lambda>1$). In the former case we get a punctured disc. In the latter case we get an annulus.