State the uniformisation theorem. List without proof the Riemann surfaces which are uniformised by $\mathbb{C}_{\infty}$ and those uniformised by $\mathbb{C}$.

Let $U$ be a domain in $\mathbb{C}$ whose complement consists of more than one point. Deduce that $U$ is uniformised by the open unit disk.

Let $R$ be a compact Riemann surface of genus $g$ and $P_{1}, \ldots, P_{n}$ be distinct points of $R$. Show that $R \backslash\left\{P_{1}, \ldots, P_{n}\right\}$ is uniformised by the open unit disk if and only if $2 g-2+n>0$, and by $\mathbb{C}$ if and only if $2 g-2+n=0$ or $-1$.

Let $\Lambda$ be a lattice and $X=\mathbb{C} / \Lambda$ a complex torus. Show that an analytic map $f: \mathbb{C} \rightarrow X$ is either surjective or constant.

Give with proof an example of a pair of Riemann surfaces which are homeomorphic but not conformally equivalent.