Spherical, parabolic and hyperbolic universal-covering types

Let $X$ be a connected Riemann surface (Riemann surfaces and holomorphic atlases); assume the Axiom of Choice (The Axiom of Choice) throughout.

The holomorphic universal cover. A universal covering space of $X$ is a covering map $p:\tilde{X}\to X$ with $\tilde{X}$ simply connected (Universal covering spaces, Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings, Simply connected topological spaces).

Existence. A Riemann surface is a topological $2$-manifold without boundary, hence locally compact and locally path connected, and every point has a neighbourhood basis of path-connected open sets (Topological manifolds are locally compact and locally path connected); being connected and locally path connected, $X$ is path connected (A connected, locally path-connected space is path-connected, because its path components are open). It is semilocally simply connected in the sense of Semilocally simply connected spaces with explicit basepoint convention: given $x\in X$ and a chart whose image is an open disc, the chart domain $U$ is homeomorphic to a disc, a nonempty convex subset of ${\mathbb{R}}^2$, hence simply connected (Every nonempty convex subset of ${\mathbb{R}}^n$ is simply connected), so every loop in $U$ is null in $U$, therefore null in $X$, and the inclusion induces the trivial map on fundamental groups. The existence theorem (Every nonempty path-connected locally path-connected semilocally simply connected space has a universal cover) therefore supplies at least one universal covering space $p:\tilde{X}\to X$.

Holomorphic structure. By A universal covering of a Riemann surface inherits a unique complex structure (which uses Countable Choice, a consequence of the Axiom of Choice) the topological universal cover carries a complex structure, unique for which the projection is a holomorphic unbranched covering, and with it $\tilde{X}$ is a second-countable Riemann surface and every deck transformation is biholomorphic. The holomorphic universal cover of $X$ means $\tilde{X}$ with this structure.

The type. The simply connected Riemann surface $\tilde{X}$ is biholomorphic to exactly one of the Riemann sphere $\widehat{\mathbb{C}}$, the complex plane $\mathbb{C}$ and the unit disc $\mathbb{D}$ (Uniformization of simply connected Riemann surfaces). Accordingly $X$ is

• spherical when its holomorphic universal cover is biholomorphic to $\widehat{\mathbb{C}}$,
• parabolic when its holomorphic universal cover is biholomorphic to $\mathbb{C}$,
• hyperbolic when its holomorphic universal cover is biholomorphic to $\mathbb{D}$.

Well-definedness. The label is independent of the universal cover chosen. Let $p:\tilde{X}\to X$ and $q:\tilde{Y}\to X$ be two universal covers and fix basepoints over the same point of $X$. By For a path-connected locally path-connected base, a universal cover maps uniquely over the base to every connected covering, and any two universal covers are uniquely isomorphic there is a unique continuous map $\phi :\tilde{X}\to \tilde{Y}$ over $X$ (that is, $q\circ \phi =p$), and it is a homeomorphism, since the same statement applied with the roles exchanged produces its inverse. The lifted complex structures of A universal covering of a Riemann surface inherits a unique complex structure make $p$ and $q$ holomorphic unbranched coverings, hence local biholomorphisms; on a small open set of $\tilde{X}$ on which $q$ is injective the map $\phi$ equals the composite of $p$ with the holomorphic local inverse of $q$, so $\phi$ is holomorphic, and symmetrically for ${\phi }^{-1}$. Thus $\phi$ is a biholomorphism (Biholomorphic maps between complex domains), the two universal covers are biholomorphic, and they determine the same one of the three models. Exactly one label occurs, for two reasons: the uniformization theorem presents $X$'s cover as one of the three models, and no two of the three are biholomorphic (The sphere, plane and disc are pairwise biholomorphically distinct), so two labels cannot both apply. The three labels are therefore exhaustive and mutually exclusive for connected Riemann surfaces.

Terminology. This is the geometric classification by the universal cover. It is not the potential-theoretic Greenian/parabolic/exceptional vocabulary of harmonic function theory; a Riemann surface that is hyperbolic in the present sense need not be Greenian, and no implication between the two classifications is asserted or used on this page. The word "parabolic" here records that the plane is the universal cover, nothing more.

Choice accounting. The definition itself records a property of $X$: the existence of the cover uses only the manifold structure of $X$, while Countable Choice enters through the lifted holomorphic structure (A universal covering of a Riemann surface inherits a unique complex structure) and full choice through the uniformization theorem (Uniformization of simply connected Riemann surfaces). No further selection is made: the type is defined by the biholomorphism class of a cover that is proved independent of the choice of cover.