庞加莱度量

黎曼曲面上的度量概要

$ds^2=\lambda^2(z,\overline{z})\, dzd\overline{z}$

$l(\gamma)=\int_\gamma \lambda(z,\overline{z})\, |dz| .$

$\mbox{Area}(M)=\int_M \lambda^2 (z,\overline{z})\,\frac{i}{2}dz \wedge d\overline{z},$

$dz \wedge d\overline{z}=(dx+i\,dy)\wedge (dx-idy)= -2i\,dx\wedge dy.$

$4\frac{\partial}{\partial z} \frac{\partial}{\partial \overline{z}} \Phi(z,\overline{z})=\lambda^2(z,\overline{z}).$
$\Delta = \frac{4}{\lambda^2} \frac {\partial}{\partial z} \frac {\partial}{\partial \overline{z}} = \frac{1}{\lambda^2} \left( \frac {\partial^2}{\partial x^2} + \frac {\partial^2}{\partial y^2} \right).$

$K=-\Delta \log \lambda ,$

$f:S\to T\,$

$\mu^2(w,\overline{w}) \; \frac {\partial w}{\partial z} \frac {\partial \overline {w}} {\partial \overline {z}} = \lambda^2 (z, \overline {z}) .$

$w(z,\overline{z})=w(z),$

$\frac{\partial}{\partial \overline{z}} w(z) = 0.$

庞加莱平面上的度量与体积元

$ds^2=\frac{dx^2+dy^2}{y^2}=\frac{dzd\overline{z}}{y^2},$

$z'=x'+iy'=\frac{az+b}{cz+d},$

$ad-bc=1$，则我们可算得

$x'=\frac{ac(x^2+y^2)+x(ad+bc)+bd}{|cz+d|^2},$

$y'=\frac{y}{|cz+d|^2},$

$dz'=\frac{dz}{(cz+d)^2},$

$dz'd\overline{z}' = \frac{dz\,d\overline{z}}{|cz+d|^4}.$

$d\mu=\frac{dx\,dy}{y^2}.$

$z_1,z_2 \in \mathbb{H}$ 度量为

$\rho(z_1,z_2)=2\tanh^{-1}\frac{|z_1-z_2|}{|z_1-\overline{z_2}|},$
$\rho(z_1,z_2)=\log\frac{|z_1-\overline{z_2}|+|z_1-z_2|}{|z_1-\overline{z_2}|-|z_1-z_2|}.$

$(z_1,z_2; z_3,z_4) = \frac{(z_1-z_2)(z_3-z_4)}{(z_2-z_3)(z_4-z_1)}.$

$\rho(z_1,z_2)= \ln (z_1,z_2^\times ; z_2, z_1^\times).$

从平面到圆盘的共形映射

$w=e^{i\phi}\frac{z-z_0}{z-\overline {z_0}} ,$

$w=\frac{iz+1}{z+i}$

i 映为圆盘的中心，0 映为圆盘的最低点。

庞加莱圆盘上的度量与体积元素

$ds^2=\frac{dx^2+dy^2}{(1-(x^2+y^2))^2}=\frac{dz\,d\overline{z}}{(1-|z|^2)^2}.$

$d\mu=\frac{dx\,dy}{(1-(x^2+y^2))^2}=\frac{dx\,dy}{(1-|z|^2)^2}.$

$z_1,z_2 \in U$ 的庞加莱度量为

$\rho(z_1,z_2)=2\tanh^{-1}\left|\frac{z_1-z_2}{1-z_1\overline{z_2}}\right| .$

穿孔圆盘模型

$q=exp(i\pi\tau)$

$ds^2=\frac{4}{|q|^2 (\log |q|^2)^2} dq d\overline{q},$

$\Phi(q,\overline{q})=4 \log \log |q|^{-2}.$

引用

• Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4.
• Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X (See Section 2.3).
• Svetlana Katok, Fuchsian Groups (1992), University of Chicago Press, Chicago ISBN 0-226-42583-5 (Provides a simple, easily readable introduction.)