§8.2 Conformal Metrics and Curvature

8.2 Conformal Metrics and Curvature

Let $Ω \subseteq ℂ$ be a region and let $𝜌 \in 𝐶 (Ω)$ be a positive function. The conformal metric (in the following chapters when we refer to metric we mean conformal) induced by $𝜌$ is given by

d 𝑠 = 𝜌 (𝑧) | d 𝑧 | or d 𝑠^{2} = {𝜌 (𝑧)}^{2} {| d 𝑧 |}^{2} .

The term “conformality” is explained in the previous section (note that this specific usage has little to do with holomorphy). The distance between two points $𝑧_{1}, 𝑧_{2} \in Ω$ is defined as

𝑑 (𝑧_{1}, 𝑧_{2}) = inf_{𝛾 \subset Ω} \int_{𝛾} 𝜌 (𝑧) | d 𝑧 |,

where the infimum is taken over all piecewise smooth curves $𝛾$ in $Ω$ joining $𝑧_{1}$ and $𝑧_{2}$ .

A $𝐶^{2}$ metric is said to be regular. The (Gaussian) curvature of the regular metric $𝜌$ at $𝑧 \in Ω$ is defined as

𝐾_{𝜌} (𝑧) = - \frac{\nabla^{2} (log 𝜌 (𝑧))}{{𝜌 (𝑧)}^{2}}, (8.2 . 1)

where $\nabla^{2} = 𝜕^{2} / 𝜕 𝑥^{2} + 𝜕^{2} / 𝜕 𝑦^{2} = 4 𝜕^{2} / 𝜕 \overline{𝑧} 𝜕 𝑧$ is the Laplacian operator. This is the same definition as the Gaussian curvature in (8.1.6) $(8.1.6) (8.1.6)$ .

The three following metrics are of particular interest in complex differential geometry:

Perhaps the most trivial metric is the Euclidean metric (also known as the parabolic metric) on $ℂ$ , and is given by
$𝜌 = 1, d 𝑠^{2} = {| d 𝑧 |}^{2} .$
The Euclidean distance between two points $𝑧_{1}, 𝑧_{2} \in ℂ$ is
$inf_{𝛾} \int_{𝛾} | d 𝑧 | = | 𝑧_{2} - 𝑧_{1} |$
is the length of the straight line segment connecting $𝑧_{1}$ and $𝑧_{2}$ . The group formed by all transformations in the form of $𝑧 \mapsto e^{i 𝜃} 𝑧 + 𝑎$ (where $𝑎 \in ℂ$ and $𝜃 \in ℝ$ ) is known as the group of rigid motions, or more abstractly, the special Euclidean group of order $2$ , denoted by $SE (2) < Aut (ℂ)$ , intuitively consists of all rotations and translations and their compositions, while the Euclidean group $E (2) > SE (2)$ consists of reflections in the form of $𝑧 \mapsto e^{i 𝜃} \overline{𝑧} + 𝑎$ . Obviously, the Euclidean metric is invariant under both groups.
From (8.2.1) $(8.2.1) (8.2.1)$ , we find that Euclidean metric has curvature $𝐾 = 0$ .
The Poincaré metric (also referred to as the hyperbolic metric) on $𝔻$ is given by
$𝜆 (𝑧) = \frac{2}{1 - {| 𝑧 |}^{2}}, d 𝑠_{𝜆}^{2} = \frac{4 {| d 𝑧 |}^{2}}{{(1 - {| 𝑧 |}^{2})}^{2}} . (8.2 . 2)$
In Lemma 3.5.4 $Lemma 3.5.4 Lemma 3.5.4$ , it was shown that the metric is invariant under $Aut (𝔻)$ .
We will now calculate the Poincaré distance between two points $𝑧_{1}, 𝑧_{2} \in ℂ$ . First assume the case where $𝑧_{1} = 0$ and $𝑧_{2} = 𝑅 \in (0, 1)$ . Consider a piecewise smooth curve $𝛾 \subset 𝔻$ parameterized by $𝑧 (𝑡)$ connecting $𝑧_{1}$ and $𝑧_{2}$ ; or in other words
$𝑧 (𝑡) = 𝑥 (𝑡) + i 𝑦 (𝑡), 𝑧 (0) = 𝑧_{1} = 0, 𝑧 (1) = 𝑧_{2} = 𝑅,$
where $𝑥 \in 𝐶^{1} ([0, 1])$ and $𝑦 \in 𝐶^{1} ([0, 1])$ are real-valued functions. Then
$\begin{aligned} \int_{𝛾} d 𝑠 & = \int_{0}^{1} \frac{2 \sqrt{𝑥^{'} {(𝑡)}^{2} + 𝑦^{'} {(𝑡)}^{2}}}{1 - {𝑥 (𝑡)}^{2} - {𝑦 (𝑡)}^{2}} d 𝑡 \\ \geq \int_{0}^{1} \frac{2 | 𝑥^{'} (𝑡) |}{1 - {𝑥 (𝑡)}^{2}} d 𝑡 \geq | \int_{0}^{1} \frac{2 𝑥^{'} (𝑡)}{1 - {𝑥 (𝑡)}^{2}} d 𝑡 | \\ = | \int_{0}^{𝑅} \frac{2 d 𝑥}{1 - 𝑥^{2}} | = log (\frac{1 + 𝑅}{1 - 𝑅}) . \end{aligned}$
Assuming that $𝛾$ is in the form of $𝑧 (𝑡) = 𝑅 𝑡, 𝑧^{'} (𝑡) = 𝑅$ where $𝑡 \in [0, 1]$ , we have
$\int_{𝛾} d 𝑠 = \int_{0}^{1} \frac{2 𝑅 d 𝑡}{1 - 𝑅^{2} 𝑡^{2}} = log (\frac{1 + 𝑅}{1 - 𝑅}) .$
Hence, the Poincaré distance between $0$ and $𝑅$ is given by
$𝑑 (0, 𝑅) = log (\frac{1 + 𝑅}{1 - 𝑅})$
and the straight line segment connecting the two points is a geodesic (path of least length under a metric or other criteria). For fixed $𝜃 \in ℝ$ since $𝑧 \mapsto 𝑧 e^{i 𝜃} \in Aut (𝔻)$ , by the Schwarz–Pick Lemma (Lemma 3.5.4 $Lemma 3.5.4 Lemma 3.5.4$ ), we have
$𝑑 (0, 𝑅) = 𝑑 (0, 𝑅 e^{i 𝜃}) = log (\frac{1 + 𝑅}{1 - 𝑅})$
by the invariance under $Aut (𝔻)$ . Now let $𝑧_{1}$ and $𝑧_{2}$ be arbitrary points in $𝔻$ . The Möbius transformation
$𝜑_{𝑧_{1}} (𝑧) = \frac{𝑧 - 𝑧_{1}}{1 - \overline{𝑧_{1}} 𝑧}$
maps $𝑧_{1}$ to $0$ and maps $𝑧_{2}$ to $\frac{𝑧_{2} - 𝑧_{1}}{1 - \overline{𝑧_{1}} 𝑧_{2}}$ . Hence, we have
$𝑑 (𝑧_{1}, 𝑧_{2}) = 𝑑 (0, \frac{𝑧_{2} - 𝑧_{1}}{1 - \overline{𝑧_{1}} 𝑧_{2}}) = log [\frac{1 + | \frac{𝑧_{2} - 𝑧_{1}}{1 - \overline{𝑧_{1}} 𝑧_{2}} |}{1 - | \frac{𝑧_{2} - 𝑧_{1}}{1 - \overline{𝑧_{1}} 𝑧_{2}} |}] = inf_{𝛾} \int_{𝛾} d 𝑠,$
which is the Poincaré distance (or hyperbolic distance) between $𝑧_{1}$ and $𝑧_{2}$ . The infimum is attained along the geodesic curve $𝛾$ parameterized by
$𝑧 (𝑡) = 𝜑_{𝑧_{1}}^{- 1} (\frac{𝑧_{2} - 𝑧_{1}}{1 - \overline{𝑧_{1}} 𝑧_{2}} 𝑡)$
for $𝑡 \in [0, 1]$ . By Theorem 5.1.8 $Theorem 5.1.8 Theorem 5.1.8$ , the geodesic is either an arc or a straight line segment passing through $𝑧_{1}$ and $𝑧_{2}$ . Since $𝜕 𝔻$ is orthogonal to the straight line passing through $0$ and $\frac{𝑧_{2} - 𝑧_{1}}{1 - \overline{𝑧_{1}} 𝑧_{2}}$ , by the conformality of $𝜑_{𝑧_{1}}^{- 1}$ , $𝜑_{𝑧_{1}}^{- 1} (𝜕 𝔻) = 𝜕 𝔻$ is orthogonal to the circular (or straight line) extension of the geodesic curve.
As a consequence of the Schwarz–Pick Lemma (Lemma 3.5.4 $Lemma 3.5.4 Lemma 3.5.4$ ), for any $𝑓 : 𝔻 \to 𝔻$ is holomorphic, we have
$𝑑 (𝑓 (𝑧_{1}), 𝑓 (𝑧_{2})) \leq 𝑑 (𝑧_{1}, 𝑧_{2}),$
where equality is attained iff $𝑓 \in Aut (𝔻)$ . The Poincaré metric has constant negative curvature $- 1$ since
$\begin{aligned} 𝐾_{𝜆} & = - \frac{4}{𝜆^{2}} \frac{𝜕^{2} log \circ 𝜆}{𝜕 \overline{𝑧} 𝜕 𝑧} = - \frac{4}{𝜆^{2}} \frac{𝜕}{𝜕 \overline{𝑧}} (\frac{𝜆_{𝑧}^{'}}{𝜆}) = - \frac{2}{𝜆^{2}} \frac{𝜕}{𝜕 \overline{𝑧}} (\frac{2 \overline{𝑧}}{1 - {| 𝑧 |}^{2}}) \\ = - \frac{2}{𝜆^{2}} (𝜆 + \overline{𝑧} 𝜆_{\overline{𝑧}}^{'}) = - \frac{{(1 - {| 𝑧 |}^{2})}^{2}}{2} (𝜆 + \overline{𝑧} \frac{2 𝑧}{{(1 - {| 𝑧 |}^{2})}^{2}}) \\ = - (1 - {| 𝑧 |}^{2}) - {| 𝑧 |}^{2} = - 1, \end{aligned}$
where $𝜆_{𝑧}^{'} = 𝜕 𝜆 / 𝜕 𝑧$ and $𝜆_{\overline{𝑧}}^{'} = 𝜕 𝜆 / 𝜕 \overline{𝑧}$ .
The spherical metric (also referred to as the elliptic metric) on $\hat{ℂ}$ is given by
$𝜎 (𝑧) = \frac{2}{1 + {| 𝑧 |}^{2}}, d 𝑠_{𝜎}^{2} = \frac{4 {| d 𝑧 |}^{2}}{{(1 + {| 𝑧 |}^{2})}^{2}} . (8.2 . 3)$
Under the inverse stereographic projection of $𝑆^{2} \to \hat{ℂ}$ , for a given $𝑧 \in \hat{ℂ}$ , the corresponding point in $𝑆^{2}$ is
$(𝑥_{1}, 𝑥_{2}, 𝑥_{3}) = (\frac{𝑧 + \overline{𝑧}}{{| 𝑧 |}^{2} + 1}, \frac{𝑧 - \overline{𝑧}}{i {| 𝑧 |}^{2} + i}, \frac{{| 𝑧 |}^{2} - 1}{{| 𝑧 |}^{2} + 1}) .$
If we let $𝑃 = (𝑥_{1}, 𝑥_{2}, 𝑥_{3})$ and $𝑄 = (\tilde{𝑥_{1}}, \tilde{𝑥_{2}}, \tilde{𝑥_{3}})$ be two points in $𝑆^{2}$ , the distance between the two points is the length of the shortest arc $\overset{⏜}{𝑥}$ (a subset of great circle passing the two points). By considering $𝑃$ and $𝑄$ as vectors from $(0, 0, 0)$ , this distance is equal to
$\begin{aligned} arccos (𝑃 \cdot 𝑄) = 2 arctan \sqrt{\frac{1 - 𝑥_{1} \tilde{𝑥_{1}} - 𝑥_{2} \tilde{𝑥_{2}} - 𝑥_{3} \tilde{𝑥_{3}}}{1 + 𝑥_{1} \tilde{𝑥_{1}} + 𝑥_{2} \tilde{𝑥_{2}} + 𝑥_{3} \tilde{𝑥_{3}}}} \\ \begin{matrix} \end{matrix} = 2 arctan \sqrt{\frac{1 - \frac{(𝑧 + \overline{𝑧}) (\tilde{𝑧} + \overline{\tilde{𝑧}})}{({| 𝑧 |}^{2} + 1) ({| \tilde{𝑧} |}^{2} + 1)} + \frac{(𝑧 - \overline{𝑧}) (\tilde{𝑧} - \overline{\tilde{𝑧}})}{({| 𝑧 |}^{2} + 1) ({| \tilde{𝑧} |}^{2} + 1)} - \frac{({| 𝑧 |}^{2} - 1) ({| \tilde{𝑧} |}^{2} - 1)}{({| 𝑧 |}^{2} + 1) ({| \tilde{𝑧} |}^{2} + 1)}}{1 + \frac{(𝑧 + \overline{𝑧}) (\tilde{𝑧} + \overline{\tilde{𝑧}})}{({| 𝑧 |}^{2} + 1) ({| \tilde{𝑧} |}^{2} + 1)} - \frac{(𝑧 - \overline{𝑧}) (\tilde{𝑧} - \overline{\tilde{𝑧}})}{({| 𝑧 |}^{2} + 1) ({| \tilde{𝑧} |}^{2} + 1)} + \frac{({| 𝑧 |}^{2} - 1) ({| \tilde{𝑧} |}^{2} - 1)}{({| 𝑧 |}^{2} + 1) ({| \tilde{𝑧} |}^{2} + 1)}}} \\ \begin{matrix} \end{matrix} = 2 arctan \sqrt{\frac{- 𝑧 \overline{\tilde{𝑧}} - \overline{𝑧} \tilde{𝑧} + {| 𝑧 |}^{2} + {| \tilde{𝑧} |}^{2}}{𝑧 \overline{\tilde{𝑧}} + \overline{𝑧} \tilde{𝑧} + {| 𝑧 |}^{2} {| \tilde{𝑧} |}^{2} + 1}} = 2 arctan \sqrt{\frac{(𝑧 - \tilde{𝑧}) (\overline{𝑧} - \overline{\tilde{𝑧}})}{(𝑧 \overline{\tilde{𝑧}} + 1) (\overline{𝑧} \tilde{𝑧} + 1)}} . \end{aligned}$
Notice that the fraction within the square root is a product between a complex number and its conjugate. Thus, this distance is equal to
$𝑑 (𝑧, \tilde{𝑧}) = 2 arctan | \frac{𝑧 - \tilde{𝑧}}{𝑧 \overline{\tilde{𝑧}} + 1} |$
in the extended complex plane. Let $\tilde{𝑧} = 𝑧 + Δ 𝑧$ . It follows that
$\begin{aligned} 𝑑 (𝑧, 𝑧 + Δ 𝑧) = 2 arctan | \frac{Δ 𝑧}{{| 𝑧 |}^{2} + 𝑧 \overline{Δ 𝑧} + 1} | \\ = 2 arctan | \frac{Δ 𝑧}{{| 𝑧 |}^{2} + 1} \frac{1}{1 + \frac{𝑧 \overline{Δ 𝑧}}{{| 𝑧 |}^{2} + 1}} | \\ = 2 arctan | \frac{Δ 𝑧}{{| 𝑧 |}^{2} + 1} (1 + 𝒪︀ (Δ 𝑧)) | = 2 arctan | \frac{Δ 𝑧}{{| 𝑧 |}^{2} + 1} + 𝒪︀ (Δ 𝑧^{2}) | \\ = 2 [| \frac{Δ 𝑧}{{| 𝑧 |}^{2} + 1} + 𝒪︀ (Δ 𝑧^{2}) | + 𝒪︀ (Δ 𝑧^{3} {[\frac{1}{{| 𝑧 |}^{2} + 1} + 𝒪︀ (Δ 𝑧)]}^{3})] \\ = 2 [| \frac{Δ 𝑧}{{| 𝑧 |}^{2} + 1} + ℴ︀ (Δ 𝑧^{2}) |], \end{aligned}$
where we have taken the liberty to coalesce orders for simplification. Since
$lim_{Δ 𝑧 \to 0} | \frac{𝑑 (𝑧, 𝑧 + Δ 𝑧)}{Δ 𝑧} | = \frac{2}{{| 𝑧 |}^{2} + 1},$
the metric as defined in (8.2.3) $(8.2.3) (8.2.3)$ has a clear geometric meaning: the distance between two points $𝑧$ and $\tilde{𝑧}$ under the metric in (8.2.3) $(8.2.3) (8.2.3)$ is the shortest distance between the corresponding points in $𝑆^{2}$ , or their spherical distance.
Thus, if curve $𝛾$ joins $𝑧$ and $\tilde{𝑧}$ , we have
$𝑑 (𝑧, \tilde{𝑧}) = inf_{𝛾} \int_{𝛾} 𝜎 (𝑧) | d 𝑧 |,$
which attains its infimum when the inverse stereographic projection of $𝛾$ is a great circle of $𝑆^{2}$ . Thus, $𝜎$ is known as the spherical metric.
The corresponding curvature is given by
$\begin{aligned} 𝐾_{𝜎} & = - \frac{4}{𝜎^{2}} \frac{𝜕^{2}}{𝜕 \overline{𝑧} 𝜕 𝑧} (log 𝜎 (𝑧)) = - \frac{4}{𝜎^{2}} \frac{𝜕}{𝜕 \overline{𝑧}} (\frac{𝜎_{𝑧}^{'}}{𝜎}) = \frac{2}{𝜎^{2}} \frac{𝜕}{𝜕 \overline{𝑧}} (\frac{2 \overline{𝑧}}{1 + {| 𝑧 |}^{2}}) \\ = \frac{2}{𝜎^{2}} (𝜎 + \overline{𝑧} 𝜎_{\overline{𝑧}}^{'}) = \frac{{(1 + {| 𝑧 |}^{2})}^{2}}{2} (\frac{2}{1 + {| 𝑧 |}^{2}} - \frac{2 {| 𝑧 |}^{2}}{{(1 + {| 𝑧 |}^{2})}^{2}}) \\ = (1 + {| 𝑧 |}^{2}) - {| 𝑧 |}^{2} = 1, \end{aligned}$
where $𝜎_{𝑧}^{'} = 𝜕 𝜎 / 𝜕 𝑧$ and $𝜎_{\overline{𝑧}}^{'} = 𝜕 𝜎 / 𝜕 \overline{𝑧}$ . This can also be verified by computing the principal curvatures of the unit sphere, which are both one.

The importance of the selected regions lies in the uniformization to be mentioned in Section 13 $Section 13 Section 13$ .

Let $Ω_{1}$ and $Ω_{2}$ be two open regions in $ℂ$ such that $𝑓 : Ω_{1} \to Ω_{2}$ is univalent (implying that $𝑓^{'} \neq 0$ by Lemma 5.1.2 $Lemma 5.1.2 Lemma 5.1.2$ ). If $𝜌$ is a metric on $Ω_{2}$ , then

𝑓^{*} 𝜌 = (𝜌 \circ 𝑓) | 𝑓^{'} | (8.2 . 4)

defines a metric on $Ω_{1}$ , referred to as the metric pullback of $𝜌$ by $𝑓$ .

Curvature as defined in (8.2.1) $(8.2.1) (8.2.1)$ is invariant under pullbacks of conformal mappings, or in the case above, we now aim to show that (under assumptions of regularity)

𝐾_{𝜌} (𝑓 (𝑧)) = 𝐾_{𝑓^{*} 𝜌} (𝑧) . (8.2 . 5)

By explicit definition,

𝐾_{𝑓^{*} 𝜌} (𝑧) = - \frac{\nabla^{2} (log \circ 𝑓^{*} 𝜌 (𝑧))}{(𝑓^{*} 𝜌) {(𝑧)}^{2}} = - \frac{\nabla^{2} (log \circ 𝜌 \circ 𝑓) (𝑧) + \nabla^{2} (log | 𝑓^{'} (𝑧) |)}{(𝑓^{*} 𝜌) {(𝑧)}^{2}} .

Since $𝑓^{'} (𝑧) \neq 0$ , $log \circ | 𝑓^{'} | = ℜ𝔢 log (𝑓^{'})$ is harmonic on $Ω_{1}$ with a vanishing Laplacian. Hence,

\begin{aligned} 𝐾_{𝑓^{*} 𝜌} (𝑧) & = - \frac{\nabla^{2} (log \circ 𝜌 \circ 𝑓 (𝑧))}{{(𝜌 \circ 𝑓)}^{2} {| 𝑓^{'} |}^{2}} = - \frac{4}{{(𝜌 \circ 𝑓)}^{2} {| 𝑓^{'} |}^{2}} \frac{𝜕}{𝜕 \overline{𝑧}} (\frac{𝜕}{𝜕 𝑧} (log \circ 𝜌 \circ 𝑓 (𝑧))) \\ = - \frac{4}{{(𝜌 \circ 𝑓)}^{2} {| 𝑓^{'} |}^{2}} \frac{𝜕}{𝜕 \overline{𝑧}} (\frac{𝜕}{𝜕 𝑓} (log \circ 𝜌 \circ 𝑓) \frac{𝜕 𝑓}{𝜕 𝑧} + \frac{𝜕}{𝜕 \overline{𝑓}} (log \circ 𝜌 \circ 𝑓) \frac{𝜕 \overline{𝑓}}{𝜕 𝑧}) \\ = - \frac{4}{{(𝜌 \circ 𝑓)}^{2} {| 𝑓^{'} |}^{2}} \frac{𝜕}{𝜕 \overline{𝑧}} (\frac{𝜕 log \circ 𝜌 \circ 𝑓}{𝜕 𝑓} \frac{𝜕 𝑓}{𝜕 𝑧} + \frac{𝜕 log \circ 𝜌 \circ 𝑓}{𝜕 \overline{𝑓}} \overline{(\frac{𝜕 𝑓}{𝜕 \overline{𝑧}})}) \\ = - \frac{4}{{(𝜌 \circ 𝑓)}^{2} {| 𝑓^{'} |}^{2}} \frac{𝜕 𝑓}{𝜕 𝑧} \frac{𝜕}{𝜕 \overline{𝑧}} (\frac{𝜕 log \circ 𝜌 \circ 𝑓}{𝜕 𝑓}) \\ = - \frac{4}{{(𝜌 \circ 𝑓)}^{2} {| 𝑓^{'} |}^{2}} \frac{𝜕 𝑓}{𝜕 𝑧} (\frac{𝜕^{2} log \circ 𝜌 \circ 𝑓}{𝜕 𝑓^{2}} \frac{𝜕 𝑓}{𝜕 \overline{𝑧}} + \frac{𝜕^{2} log \circ 𝜌 \circ 𝑓}{𝜕 \overline{𝑓} 𝜕 𝑓} \overline{(\frac{𝜕 𝑓}{𝜕 𝑧})}) \\ = - \frac{4}{{(𝜌 \circ 𝑓)}^{2}} \frac{𝜕^{2} log \circ 𝜌 \circ 𝑓}{𝜕 \overline{𝑓} 𝜕 𝑓} = - \frac{\nabla_{𝑓}^{2} (log \circ 𝜌 \circ 𝑓)}{{(𝜌 \circ 𝑓)}^{2}} = 𝐾_{𝜌} (𝑓 (𝑧)) . \end{aligned}

For a given metric $d 𝑠 = 𝜆 (𝑧) | d 𝑧 |$ , if there is some other parameterization such that $d 𝑠 = 𝜆^{'} (𝑧^{'}) | {d 𝑧}^{'} |$ , $𝑧^{'} = 𝑓 (𝑧)$ is conformal, then the relation is given by $𝜆 = 𝑓^{*} 𝜆^{'}$ . Under differing parameterizations of a metric $d 𝑠$ , we once again have the invariance of curvature.

Remark.

In the work of Ahlfors and others, there is no

2 \cdot

term in front of the hyperbolic and spherical metrics, in which cases the curvatures are

- 4

and

4

respectively.