# 庞加莱度量

## 黎曼曲面上的度量概要

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

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

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

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

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

${\displaystyle K=-\Delta \log \lambda ,}$

${\displaystyle f:S\to T\,}$

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

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

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

## 庞加莱平面上的度量与体积元

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

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

${\displaystyle ad-bc=1}$ ，则我们可算得

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

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

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

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

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

${\displaystyle z_{1},z_{2}\in \mathbb {H} }$  度量为

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

${\displaystyle (z_{1},z_{2};z_{3},z_{4})={\frac {(z_{1}-z_{2})(z_{3}-z_{4})}{(z_{2}-z_{3})(z_{4}-z_{1})}}.}$

${\displaystyle \rho (z_{1},z_{2})=\ln(z_{1},z_{2}^{\times };z_{2},z_{1}^{\times }).}$

## 从平面到圆盘的共形映射

${\displaystyle w=e^{i\phi }{\frac {z-z_{0}}{z-{\overline {z_{0}}}}},}$

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

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

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

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

${\displaystyle z_{1},z_{2}\in U}$  的庞加莱度量为

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

## 穿孔圆盘模型

${\displaystyle q=exp(i\pi \tau )}$

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

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

## 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.)