Riemann Surfaces: With 27 FiguresThe present volume is the culmination often years' work separately and joint ly. The idea of writing this book began with a set of notes for a course given by one of the authors in 1970-1971 at the Hebrew University. The notes were refined serveral times and used as the basic content of courses given sub sequently by each of the authors at the State University of New York at Stony Brook and the Hebrew University. In this book we present the theory of Riemann surfaces and its many dif ferent facets. We begin from the most elementary aspects and try to bring the reader up to the frontier of present-day research. We treat both open and closed surfaces in this book, but our main emphasis is on the compact case. In fact, Chapters III, V, VI, and VII deal exclusively with compact surfaces. Chapters I and II are preparatory, and Chapter IV deals with uniformization. All works on Riemann surfaces go back to the fundamental results of Rie mann, Jacobi, Abel, Weierstrass, etc. Our book is no exception. In addition to our debt to these mathematicians of a previous era, the present work has been influenced by many contemporary mathematicians. |
From inside the book
Results 1-3 of 62
Page 63
... conclude that Sou B5x = SSu 5 ; ^ 5x- - — i ( 5,5x ) = So , 5x — So2 5 ; = −2i ( Im πjk ) ( as usual л is the ( j , k ) -entry of the matrix II ) . In particular , πjk Applying the same argument to Ꮎ we conclude that Im л¡¡ > 0 . πjj ...
... conclude that Sou B5x = SSu 5 ; ^ 5x- - — i ( 5,5x ) = So , 5x — So2 5 ; = −2i ( Im πjk ) ( as usual л is the ( j , k ) -entry of the matrix II ) . In particular , πjk Applying the same argument to Ꮎ we conclude that Im л¡¡ > 0 . πjj ...
Page 260
... conclude that ( using Riemann - Hurwitz ) · deg D ; + 1 − ğ = ğ − 1 + -Σ Σ ( n − = D ğ − In We compute next g = g - X1 1 XI - limus ) ( 2.5.4 ) n l \ n m = 1 12mlj n l❘n m = 1 = 0-1 + Σ - ΙΣ Σ n - 1 Σ . λmlj j = 1 = Recall that 7ml ...
... conclude that ( using Riemann - Hurwitz ) · deg D ; + 1 − ğ = ğ − 1 + -Σ Σ ( n − = D ğ − In We compute next g = g - X1 1 XI - limus ) ( 2.5.4 ) n l \ n m = 1 12mlj n l❘n m = 1 = 0-1 + Σ - ΙΣ Σ n - 1 Σ . λmlj j = 1 = Recall that 7ml ...
Page 321
... conclude that ( multiply the above function by w ) - Z1 = P is a canonical divisor on M. Pty - 1X2 ( X2 ) ( 1 ) ... ( X2 ) ( N - 1 ) ( 3.6.9 ) In case ( 3.6.8 ) we choose P1 e M as the base point of the map 9 : M → J ( M ) . We conclude ...
... conclude that ( multiply the above function by w ) - Z1 = P is a canonical divisor on M. Pty - 1X2 ( X2 ) ( 1 ) ... ( X2 ) ( N - 1 ) ( 3.6.9 ) In case ( 3.6.8 ) we choose P1 e M as the base point of the map 9 : M → J ( M ) . We conclude ...
Other editions - View all
Common terms and phrases
a₁ A₂ abelian differentials analytic arbitrary assume automorphism b₁ base point canonical homology basis Choose compact Riemann surface compact surface component compute conclude conformal disc conformally equivalent coordinate Corollary curve D₁ defined denote Dirichlet problem divisor of degree domain elliptic equation exists f₁ finite fixed points follows Fuchsian Fuchsian group function f genus g group G H¹(M harmonic function Hence holomorphic differential holomorphic function holomorphic mapping hyperbolic hyperelliptic surfaces integral divisor Lemma m₁ matrix meromorphic function metric Möbius transformation multiplicative neighborhood non-constant ordp P₁ P₂ parabolic period matrix point of order proj PROOF Proposition puncture Q₁ Recall Remark Riemann-Hurwitz Riemann-Roch theorem subgroup subharmonic subset surface of genus theta function topological torus triangulation u₁ unique universal covering space v₁ vector w₁ Weierstrass points z₁ zero Σ Σ