This book is novel in its broad perspective on Riemann surfaces: the text systematically explores the connection with other fields of mathematics. The book can serve as an introduction to contemporary mathematics as a whole, as it develops background material from algebraic topology, differential geometry, the calculus of variations, elliptic PDE, and algebraic geometry. The book is unique among textbooks on Riemann surfaces in its inclusion of an introduction to TeichmA¼ller theory. For this new edition, the author has expanded and rewritten several sections to include additional material and to improve the presentation.We now prove the Riemann-Hurwitz formula: Theorem 2.5.2 Let f : XXI a XX be a non-constant holomorphic map of degree m ... is given in a local chart near p, by w = 2aquot;, and let B (r) be a disc of radius r around p; in this chart. Since f is a local isometry, we will have, as r -agt; 0, 1 62 - - 4 27T *i-U-1 B, (r) 62.6% log (A (*)a#39;) : dz dz 7m, 2 i - - 4 log A) - - 27T 21aU B, (r) 6wApTU ( Og ) 2 du da#39;U a m (2a2g2) by Cor.
|Title||:||Compact Riemann Surfaces|
|Publisher||:||Springer Science & Business Media - 2013-04-17|