Poincaré metric on a hyperbolic Riemann surface

Let $X$ be a connected Riemann surface (Riemann surfaces and holomorphic atlases) of hyperbolic universal-covering type, meaning that its holomorphic universal cover is biholomorphic to the unit disc $\mathbb{D}$ (Spherical, parabolic and hyperbolic universal-covering types); assume the Axiom of Choice (The Axiom of Choice) throughout, as in that definition. On $\mathbb{D}$ let

$d{s}_{\mathbb{D}}=\frac{2\mathrm{\mid }dz\mathrm{\mid }}{1-\mathrm{\mid }z{\mathrm{\mid }}^2}$

be the Poincaré length element (The Poincare metric and distance on the unit disc), with the normalisation (factor $2$) fixed there for all later surface pages.

Uniformizations. By the definition of hyperbolic type there are a holomorphic universal covering $p:\tilde{X}\to X$, with $\tilde{X}$ carrying the complex structure of A universal covering of a Riemann surface inherits a unique complex structure for which $p$ is a holomorphic unbranched covering, and a biholomorphism $\psi :\tilde{X}\to \mathbb{D}$ (Biholomorphic maps between complex domains). Call such a pair $(p,\psi )$, or briefly $\psi$, a uniformization of $X$.

The Poincaré length element. Let $V\subseteq X$ be a connected evenly covered open set with inverse sheet $s:V\to \tilde{X}$, that is, $p\circ s={\mathrm{id}}_V$ (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings). Then $s$ is holomorphic and $\psi \circ s:V\to \mathbb{D}$ is a holomorphic local biholomorphism, and on $V$ define

$d{s}_X{\mid }_V:=(\psi \circ s{)}^{\ast }d{s}_{\mathbb{D}},$

the local pushforward of $d{s}_{\mathbb{D}}$ along the inverse sheet $s$ of the covering $p$, equivalently its pullback along $\psi \circ s$. In a holomorphic chart $z$ of $X$ with domain $V$ this reads

$d{s}_X=\frac{2\mathrm{\mid }{F}^{\mathrm{\prime }}(\zeta )\mathrm{\mid }}{1-\mathrm{\mid }F(\zeta ){\mathrm{\mid }}^2}\mathrm{\mid }d\zeta \mathrm{\mid },F:=\psi \circ s\circ z^{-1},\zeta =z(x),$

a positive smooth coefficient because $F^{\mathrm{\prime }}$ does not vanish. The Poincaré length element, also called the Poincaré metric, of $X$ is the conformal metric obtained by patching these local expressions.

Consistency. The patching requires three checks; the definition is local, so it suffices to compare the expressions where both are defined.

1. Charts. If $z^{\mathrm{\prime }}$ is a further holomorphic chart on $V$, then with $\varphi :=z\circ z^{\mathrm{\prime }-1}$ one has $F\circ \varphi =\psi \circ s\circ z^{\mathrm{\prime }-1}$ and, by the chain rule, $\frac{2\mathrm{\mid }(F\circ \varphi {)}^{\mathrm{\prime }}({\zeta }^{\mathrm{\prime }})\mathrm{\mid }}{1-\mathrm{\mid }(F\circ \varphi )({\zeta }^{\mathrm{\prime }}){\mathrm{\mid }}^2}=\frac{2\mathrm{\mid }{F}^{\mathrm{\prime }}(\varphi ({\zeta }^{\mathrm{\prime }}))\mathrm{\mid }}{1-\mathrm{\mid }F(\varphi ({\zeta }^{\mathrm{\prime }})){\mathrm{\mid }}^2}\mathrm{\mid }{\varphi }^{\mathrm{\prime }}({\zeta }^{\mathrm{\prime }})\mathrm{\mid },$ which is exactly the transformation rule of a conformal metric under the coordinate change $\zeta =\varphi ({\zeta }^{\mathrm{\prime }})$. So each local expression defines a conformal metric on its chart, independent of the chart.

2. Inverse sheets. Let $s,s^{\mathrm{\prime }}:V\to \tilde{X}$ be two inverse sheets over a connected evenly covered $V$ and let $x_0\in V$. Then $s(x_0)$ and $s^{\mathrm{\prime }}(x_0)$ lie in the same fibre $p^{-1}(x_0)$. The base $X$ is a connected manifold, hence path connected, locally path connected and semilocally simply connected, and for a universal cover the deck group acts transitively on each fibre (For a path-connected locally path-connected semilocally simply connected base, the deck group of a universal cover is isomorphic to the fundamental group): a path in the simply connected total space joining two points of a fibre projects to a loop whose lifted endpoint is the second fibre point, and the loop class gives a deck transformation moving the first point to the second. Choose $h\in \mathrm{Deck}(p)$ with $h(s(x_0))=s^{\mathrm{\prime }}(x_0)$. Since $V$ is connected, $s(V)$ lies in a single sheet of $p^{-1}(V)$ and $s^{\mathrm{\prime }}(V)$ lies in a single sheet; shrinking to a connected open neighbourhood $W\subseteq V$ of $x_0$ over which $h\circ s$ is defined, both $s^{\mathrm{\prime }}$ and $h\circ s$ are continuous inverses of $p$ on $W$ with values in the same sheet, where $p$ is injective, so $s^{\mathrm{\prime }}=h\circ s$ on $W$. By A universal covering of a Riemann surface inherits a unique complex structure the deck transformation $h$ is biholomorphic, so $h_{\mathbb{D}}:=\psi \circ h\circ {\psi }^{-1}$ is an automorphism of $\mathbb{D}$, and by The Poincare distance has the formula $2\mathrm{artanh}\mathrm{\mid }{\phi }_z(w)\mathrm{\mid }$ and is disc-automorphism invariant (whose proof derives the pointwise identity $2\mathrm{\mid }{h}_{\mathbb{D}}^{\mathrm{\prime }}(z)\mathrm{\mid }\mathrm{/}(1-\mathrm{\mid }{h}_{\mathbb{D}}(z){\mathrm{\mid }}^2)=2\mathrm{/}(1-\mathrm{\mid }z{\mathrm{\mid }}^2)$ for every disc automorphism) the length element $d{s}_{\mathbb{D}}$ is $h_{\mathbb{D}}$-invariant. Hence on $W$ $\psi \circ s^{\mathrm{\prime }}=h_{\mathbb{D}}\circ (\psi \circ s),\text{so}(\psi \circ s^{\mathrm{\prime }}{)}^{\ast }d{s}_{\mathbb{D}}=(\psi \circ s{)}^{\ast }d{s}_{\mathbb{D}},$ and the local metrics defined from $s$ and from $s^{\mathrm{\prime }}$ agree near $x_0$; as $x_0$ was arbitrary, they agree on $V$.

3. Uniformization and cover. If $(p,\psi )$ and $(p,{\psi }^{\mathrm{\prime }})$ are uniformizations of $X$ with the same cover, then ${\psi }^{\mathrm{\prime }}=h\circ \psi$ for the disc automorphism $h:={\psi }^{\mathrm{\prime }}\circ {\psi }^{-1}$, and ${\psi }^{\mathrm{\prime }}\circ s=h\circ (\psi \circ s)$; the same invariance gives the identical local metric. If instead $p^{\mathrm{\prime }}:{\tilde{X}}^{\mathrm{\prime }}\to X$ is a further holomorphic universal cover and $(p^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }})$ is a uniformization of it, then by the well-definedness discussion of Spherical, parabolic and hyperbolic universal-covering types there is a biholomorphism $\mathrm{\Phi }:{\tilde{X}}^{\mathrm{\prime }}\to \tilde{X}$ over $X$, that is, $p\circ \mathrm{\Phi }=p^{\mathrm{\prime }}$, and then $\psi \circ \mathrm{\Phi }$ is a uniformization of $p^{\mathrm{\prime }}$ while ${\mathrm{\Phi }}^{-1}\circ s$ is an inverse sheet of $p^{\mathrm{\prime }}$ over $V$ with $(\psi \circ \mathrm{\Phi })\circ ({\mathrm{\Phi }}^{-1}\circ s)=\psi \circ s$; the local expressions coincide verbatim. So the metric depends only on $X$.

Since every point of $X$ lies in a connected evenly covered open set, the consistent local conformal metrics patch to a global conformal metric $d{s}_X$ on $X$, given in each holomorphic chart by a positive smooth coefficient ${\rho }_X$.

Poincaré length and distance. Let $\gamma :[a,b]\to X$ be a piecewise $C^1$ curve (Piecewise c one curve on a manifold). Its Poincaré length is

${\mathrm{\ell }}_X(\gamma ):={\int }_a^b{\rho }_X(\gamma (t))\mathrm{\mid }{\gamma }^{\mathrm{\prime }}(t)\mathrm{\mid }dt,$

the integrand computed in the finitely many holomorphic charts covering the pieces; the value is independent of those charts and of the subdivision by the transformation rule of check 1 and the chain rule. The image of $\gamma$ is compact and ${\rho }_X$ is continuous and positive, hence bounded on it, so ${\mathrm{\ell }}_X(\gamma )<\mathrm{\infty }$. The Poincaré distance is

$d_X(x,y):=\inf \{{\mathrm{\ell }}_X(\gamma ):\gamma \text{ a piecewise }{C}^1\text{ curve in }X\text{ from }x\text{ to }y\},x,y\in X\mathrm{.}$

The set is nonempty: a Riemann surface is locally path connected, and two points are joined inside finitely many chart discs, in which the Euclidean segments straighten to a piecewise $C^1$ curve. Each such curve has finite length, so $d_X$ is finite; it is symmetric, and the triangle inequality holds by concatenating curves and adding integrals. It is positive for $x\mathrm{\ne }y$: in a chart carrying $x$ to $0$, take $r>0$ with the closed disc of radius $r$ about $0$ missing the image of $y$ and with ${\rho }_X\ge c>0$ on that disc; every curve from $x$ to $y$ leaves the disc, and the part up to the first exit has Euclidean length at least $r$, hence Poincaré length at least $cr$. Thus $d_X$ is a metric on $X$, and ${\mathrm{\ell }}_X,d_X$ are its length and distance functions.

Equivalent description. By construction the pullback of $d{s}_X$ under the covering is the pullback of the disc metric under the uniformization, $p^{\ast }d{s}_X={\psi }^{\ast }d{s}_{\mathbb{D}}$ on $\tilde{X}$; the check 2 above is exactly the statement that ${\psi }^{\ast }d{s}_{\mathbb{D}}$ is invariant under deck transformations. Indeed $h^{\ast }(p^{\ast }d{s}_X)=(p\circ h{)}^{\ast }d{s}_X=p^{\ast }d{s}_X$ for every deck transformation $h$, so deck transformations act on $\tilde{X}$ by isometries of the pulled-back metric. This is the metric statement that the later surface page develops for the action of the deck group on the disc.

Scope and normalisation. The construction is made only for surfaces of hyperbolic universal-covering type; no metric is introduced here on surfaces whose universal cover is the sphere or the plane. The normalisation is the one fixed in The Poincare metric and distance on the unit disc (factor $2$, the curvature $-1$ normalisation of the disc model), so that in a coordinate obtained from an inverse covering sheet followed by $\psi$, the surface length element is $2\mathrm{\mid }dz\mathrm{\mid }\mathrm{/}(1-\mathrm{\mid }z{\mathrm{\mid }}^2)$ on that coordinate image. An arbitrary coordinate-disc chart uses the derivative-weighted formula in the Definition. In particular, when $X=\mathbb{D}$ and $z=2w$ on $\mathrm{\mid }w\mathrm{\mid }<1\mathrm{/}2$, that formula gives $d{s}_X=\mathrm{\mid }dz\mathrm{\mid }\mathrm{/}(1-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}4)$.

Choice accounting. The definition makes no new selection: the cover and the uniformization are supplied by the type definition, and the consistency checks show that neither the choice of the cover nor the choice of the uniformization affects $d{s}_X$. The Axiom of Choice enters only through Spherical, parabolic and hyperbolic universal-covering types, namely as Countable Choice for the lifted holomorphic structure and as full choice through the uniformization theorem; every other selection above is a finite one (finitely many charts covering a curve, finitely many Euclidean segments).