Let \(\rho(z)|dz|\) be the hyperbolic metric on \[ X=\widehat{\mathbb C}\setminus\{0,1,\infty\}. \] Show that there are constants \(A,B>0\) such that, for all sufficiently small nonzero \(z\), \[ \frac{A}{|z\log|z||} \le \rho(z) \le \frac{B}{|z\log|z||}. \]
Hint: write the covering map using its period as a holomorphic map from the punctured disk and compare it with the standard hyperbolic metric on a punctured disk.
Solution
Near the puncture \(0\) one has \[ \boxed{ \rho(z)\asymp\frac1{|z\log|z||}. } \] More explicitly, there are \(A,B,\varepsilon>0\) such that \[ \frac{A}{|z\log|z||} \le \rho(z) \le \frac{B}{|z\log|z||} \qquad(0<|z|<\varepsilon). \]
1. The period \(2\) of the modular covering factors it through a punctured disk.
The universal covering \[ \lambda:\mathbb H\to X \] is invariant under the parabolic deck transformation \[ \tau\longmapsto\tau+2. \] Set \[ q=e^{\pi i\tau}. \] Then \(q(\tau+2)=q(\tau)\), and the quotient \[ \mathbb H/\langle\tau\mapsto\tau+2\rangle \] is biholomorphic to the punctured disk \(\Delta^*\). Hence there is a holomorphic covering map \[ f:\Delta^*\to X \] such that \[ \lambda(\tau)=f(e^{\pi i\tau}). \] The modular function satisfies \(\lambda(\tau)\to0\) as \(\operatorname{Im}\tau\to\infty\), so the end \(\operatorname{Im}\tau\to\infty\) maps to the puncture \(0\in\widehat{\mathbb C}\). Thus \(f\) extends holomorphically across \(q=0\) with \[ f(0)=0. \] The loop \(q=re^{2\pi it}\) represents the primitive quotient translation \(\tau\mapsto\tau+2\), which is the primitive peripheral loop about this puncture. Therefore the local degree of the extension at \(0\) is one, so \[ f'(0)\ne0. \]
2. The hyperbolic metric on \(\Delta^*\) is\[ \frac{|dq|}{|q\log|q||}. \]
Under \(q=e^{\pi i\tau}\) one has \(|dq|=\pi|q|\,|d\tau|\) and \(|\log|q||=\pi\operatorname{Im}\tau\), so this metric is the quotient of the curvature \(-1\) metric \(|d\tau|/\operatorname{Im}\tau\) on \(\mathbb H\) by \(\tau\mapsto\tau+2\). Taking \(\rho(z)|dz|\) with curvature \(-1\) as well, the holomorphic covering \(f:\Delta^*\to X\) is a local isometry. Therefore \[ f^*\bigl(\rho(z)|dz|\bigr) =\frac{|dq|}{|q\log|q||}. \] Equivalently, \[ \rho(f(q))\,|f'(q)| =\frac1{|q\log|q||}. \]
3. The local coordinate \(f\) compares \(q\) and \(z=f(q)\) up to fixed constants.
Write \[ a=f'(0)\ne0. \] Then \[ f(q)=aq+O(q^2), \qquad f'(q)=a+O(q). \] Hence, after shrinking to \(0<|q|<\delta\), there are constants \(c_1,c_2,c_3,c_4>0\) such that \[ c_1|q|\le |f(q)|\le c_2|q|, \qquad c_3\le |f'(q)|\le c_4. \] Moreover \[ \log|f(q)| =\log|q|+\log|a+O(q)|. \] The second term is bounded as \(q\to0\), whereas \(\log|q|\to-\infty\). Thus, after shrinking \(\delta\) again, there are constants \(c_5,c_6>0\) with \[ c_5|\log|q|| \le |\log|f(q)|| \le c_6|\log|q||. \] Since \(f'(0)\ne0\), the inverse function theorem also shows that \(f\) maps a small disk about \(0\) biholomorphically onto a neighborhood of \(0\). Hence every sufficiently small nonzero \(z\) can be written uniquely as \(z=f(q)\) with \(q\) small.
4. Substitute these comparisons into the local-isometry identity.
From 2, \[ \rho(f(q)) =\frac1{|f'(q)|\,|q|\,|\log|q||}. \] The estimates of 3 show that the denominator on the right is comparable, by positive constants independent of \(q\), to \[ |f(q)|\,|\log|f(q)||. \] Thus there are \(A,B>0\) such that \[ \frac{A}{|f(q)\log|f(q)||} \le \rho(f(q)) \le \frac{B}{|f(q)\log|f(q)||} \] for all sufficiently small nonzero \(q\). Writing \(z=f(q)\) gives \[ \boxed{ \frac{A}{|z\log|z||} \le \rho(z) \le \frac{B}{|z\log|z||} } \] for all sufficiently small nonzero \(z\).