报告题目： Uniformisation and description of a once-punctured annulus

报告人： 张坦然博士（日本东北大学 ）

报告时间：  2013年6月19日下午3:00-4:00

报告地点：　数学楼二楼学术报告厅

摘要： The Uniformisation Theorem shows that the universal covering space $tilde{X}$ of an arbitrary Riemann surface X is homeomorphic, by a conformal map $m$, to either the Riemann sphere $hat{C}$, the complex plane $C$ or the unit disk $D$. And then the fundamental group  $Pi_1(X)$  has a representation as a group $G$ of conformal homeomorphisms of  $m(tilde{X})$. This theorem also indicates that if $tilde{X}$ is homeomorphic to a proper subset of C with at least three boundary points, then $tilde{X}$ is conformally equivalent to a quotient space $D/G$, where $G$ is a torsion-free Fuchsian group that acts (discontinuously) on $D$ (or $H$). The group $G$ is isomorphic to  $Pi_1(X)$. Hempel and Smith studied the hyperbolic Riemann surface model of the twice-punctured disk $Dbackslash {p_1; p_2}$ in 1980s. They estimated the hyperbolic density on it near a once-puncture and considered the coalescing of the two punctures. Later on Beardon gave five different ways to uniformize $Dbackslash {p_1; p_2}$ in 2012. He investigated several conformal invariants to characterize $Dbackslash {p_1; p_2}$ considering the fundamental domain, symmetric collars and extremal length. We extend his work to the once-punctured annulus  $A := {z : 1/R < |z| < R}backslash {a}, R > 1, 1/R < a < R$. We provide several parameter pairs to uniformize and characterize it. The main tools we use are Mobius transformations, covering space, homotopy classes and elliptic integrals.

References:

[1] A.F. Beardon, On the geometry of discrete groups, Graduate Texts in Mathematics, no. 91, Springer-Verlag, 1983.

[2] A.F. Beardon, The uniformisation of a twice-punctured disc, Comput. Methods Funct. Theory 12 (2012), no. 2, 585-596.

[3] J.A. Hempel and S.J. Smith, Uniformization of the twice-punctured disc-problems of confluence, Bull. Australian Math. Soc. 39 (1989), 369-387.