Classifying Annuli up to Holomorphic Isomorphism

An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/

Goals: To show that the various annuli with inner radii in the real open unit interval and with outer radius unity are all non-isomorphic as Riemann surfaces


Keywords: Upper half-plane, universal covering, fundamental group, deck transformation group, Moebius transformations, real special linear group, real projective (special) linear group, simply connected, biholomorphic map, holomorphic isomorphism, infinite cyclic group, parabolic Moebius transformation, hyperbolic Moebius transformation, fixed point, extended plane, abelian fundamental group, commuting Moebius transformations, commuting deck transformations, punctured unit disc, annulus, unique lifting property