Download Compact Riemann surfaces by Bobenko A. PDF

By Bobenko A.

Show description

Read Online or Download Compact Riemann surfaces PDF

Similar calculus books

Mathematical problems of control theory: an introduction

Exhibits in actual fact how the research of concrete regulate platforms has prompted the improvement of the mathematical instruments wanted for fixing such difficulties. The Aizerman and Brockett difficulties are mentioned and an advent to the idea of discrete keep watch over platforms is given.

Von Karman evolution equations: Well-posedness and long time dynamics

The most aim of this booklet is to debate and current effects on well-posedness, regularity and long-time habit of non-linear dynamic plate (shell) versions defined through von Karman evolutions. whereas the various effects awarded listed below are the outgrowth of very fresh reports through the authors, together with a few new unique effects right here in print for the 1st time authors have supplied a finished and fairly self-contained exposition of the final subject defined above.

Distributions, Sobolev spaces, elliptic equations

It's the major objective of this booklet to enhance at an obtainable, reasonable point an $L_2$ concept for elliptic differential operators of moment order on bounded delicate domain names in Euclidean n-space, together with a priori estimates for boundary-value difficulties by way of (fractional) Sobolev areas on domain names and on their limitations, including a similar spectral conception.

Introduction to the Theory and Application of the Laplace Transformation

In anglo-american literature there exist various books, dedicated to the applying of the Laplace transformation in technical domain names equivalent to electrotechnics, mechanics and so on. mainly, they deal with difficulties which, in mathematical language, are ruled via ordi­ nary and partial differential equations, in numerous bodily dressed types.

Extra resources for Compact Riemann surfaces

Example text

7 Let P0 be a Weierstrass point on R and z a local parameter at P0 , with z(P0 ) = 0. The order τ (P0 ) of the zero of ∆ at P0 ∆ = z τ (P0 ) O(1) (96) is called the weight of the Weierstrass point P0 . It turnes out that ∆ is well defined on R globally. 8 If to every local coordinate z : U ⊂ R → V ⊂ C there assigned a holomorphic function r(z) such that r = r(z)dz q , q∈Z (97) is invariant under holomorphic coordinate changes (49) one says that the holomorphic q-differential r is defined on R. In the same way as for the Abelian differentials one defines the divisor (r) of the qdifferentials.

G. 6 The differentials ΩR , ΩRQ with the singularities (63), (64) and all zero a-periods (67) are called the normalized Abelian differentials of the second and third kind. 12 Given a compact Riemann surface R with a canonical basis of cycles a1 , b1 , . . , ag , bg , points R, Q ∈ R, a local parameter z at R and N ∈ N there exist unique (N ) normalized Abelian differentials of the second ΩR and of the third ΩRQ kind. 4. The proof of the uniqueness is simple. 6. 7 Abelian differentials of the second and third kind can be normalized by a more symmetric then (67) condition.

In Fig. 20 the parts of the cycles lying on the ”lower” sheet of the covering are marked by dotted lines. b1 a1 Πg ag bg Figure 19: Canonical basis of cycles on the planar model Πg of compact Riemann surface. b1 b2 λ1 λ2 a1 λ3 λ4 a2 bg λ2g−1 λ2g λ2g+1 a3 Figure 20: Canonical basis of cycles of a hyperelliptic Riemann surface. λ2g+2 4 ABELIAN DIFFERENTIALS 4 32 Abelian differentials Our main goal is to construct functions on compact Riemann surfaces with prescribed analytical properties (for example, meromorphic functions with prescribed singularities).

Download PDF sample

Rated 5.00 of 5 – based on 22 votes