Riemann SurfacesIt is gratifying to learn that there is new life in an old field that has been at the center of one's existence for over a quarter of a century. It is particularly pleasing that the subject of Riemann surfaces has attracted the attention of a new generation of mathematicians from (newly) adjacent fields (for example, those interested in hyperbolic manifolds and iterations of rational maps) and young physicists who have been convinced (certainly not by mathematicians) that compact Riemann surfaces may play an important role in their (string) universe. We hope that non-mathematicians as well as mathematicians (working in nearby areas to the central topic of this book) will also learn part of this subject for the sheer beauty and elegance of the material (work of Weierstrass, Jacobi, Riemann, Hilbert, Weyl) and as healthy exposure to the way (some) mathematicians write about mathematics. We had intended a more comprehensive revision, including a fuller treatment of moduli problems and theta functions. Pressure of other commitments would have substantially delayed (by years) the appearance of the book we wanted to produce. We have chosen instead to make a few modest additions and to correct a number of errors. We are grateful to the readers who pointed out some of our mistakes in the first edition; the responsibility for the remaining mistakes carried over from the first edition and for any new ones introduced into the second edition remains with the authors. June 1991 Jerusalem H. M. |
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a₁ A₂ Abel's theorem abelian differentials algebraic analytic arbitrary assume automorphism b₁ base point canonical homology basis Choose compact Riemann surface compact surface component compute conclude conformally equivalent consider Corollary D₁ defined denote dimension disc Div(M divisors of degree domain elliptic equation exists f₁ finite fixed points follows function ƒ genus g H¹(M harmonic function Hence holomorphic differentials holomorphic function holomorphic mapping hyperelliptic surface integral divisor involution isomorphism Jacobian variety Lemma linear m₁ meromorphic function Möbius transformation neighborhood non-constant non-gaps P₁ P₂ parabolic period matrix point of order polar divisor poles proj PROOF Proposition puncture Q₁ recall Remark Riemann-Hurwitz Riemann-Roch theorem simply connected space subgroup subharmonic surface of genus theta function topological torus unique V₁ vanish identically vector W₁ w₂ Weierstrass points zero