By Alexander I. Bobenko (auth.), Alexander I. Bobenko, Christian Klein (eds.)
This quantity deals a well-structured evaluate of existent computational techniques to Riemann surfaces and people at the moment in improvement. The authors of the contributions symbolize the teams offering publically to be had numerical codes during this box. therefore this quantity illustrates which software program instruments can be found and the way they are often utilized in perform. additionally examples for recommendations to partial differential equations and in floor idea are awarded. The meant viewers of this ebook is twofold. it may be used as a textbook for a graduate direction in numerics of Riemann surfaces, within which case the traditional undergraduate historical past, i.e., calculus and linear algebra, is needed. specifically, no wisdom of the idea of Riemann surfaces is anticipated; the mandatory heritage during this idea is inside the advent bankruptcy. even as, this e-book can also be meant for experts in geometry and mathematical physics using the idea of Riemann surfaces of their study. it's the first publication on numerics of Riemann surfaces that displays the development made during this box over the last decade, and it includes unique effects. There are more and more functions that contain the review of concrete features of versions analytically defined when it comes to Riemann surfaces. Many challenge settings and computations during this quantity are stimulated through such concrete functions in geometry and mathematical physics.
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Additional info for Computational Approach to Riemann Surfaces
Fig. 8. 2 Symmetric Riemann Surfaces as Coverings The construction of Sect. 2 can be also applied to Riemann surfaces. Theorem 7. Let R be a (compact) Riemann surface and let G be a ﬁnite group of holomorphic automorphisms1 of order |G|. Then R/G is a Riemann surface with the complex structure determined by the condition that the canonical projection π : R → R/G is holomorphic. This is an |G|-sheeted covering, ramiﬁed at the ﬁxed points of G. 1 This group is always ﬁnite if the genus ≥ 2. 1 Riemann Surfaces 17 The canonical projection π deﬁnes an |G|-sheeted covering.
The vanishing of theta functions at some points follows from their algebraic properties. Deﬁnition 32. Half-periods of the period lattice Δ = 2πiα + Bβ , α = (α1 , . . , αg ) , β = (β1 , . . , βg ), α k , βk ∈ 0, 1 2 . are called half periods or theta characteristics. A half period is called even (odd) if 4(α, β) = 4 αk βk is even (odd). We denote the theta characteristic by Δ = [α, β]. The theta function θ(z) vanishes in all odd theta characteristics θ(Δ) = θ(−Δ + 4πiα + 2Bβ) = θ(−Δ) exp(−4πi(α, β)) .
Theorem 21. Any Riemann surface of genus g = 2 is hyperelliptic. This is not diﬃcult to prove. The zero divisor (ω) of a holomorphic diﬀerential on a Riemann surface R of genus 2 is of degree 2 = 2g − 2. Since i((ω)) > 0, the Riemann–Roch theorem implies l((ω)) ≥ 2. Special divisors on hyperelliptic Riemann surfaces are characterized by the following simple property. ˆ Proposition 6. 5) with branch points λk , k = 1 . . , N . A positive divisor D of degree g is singular if and only if it contains a pair of points (μ0 , λ0 ), (−μ0 , λ0 ) with the same λ-coordinate λ0 = λk or a double branch point 2(0, λk ).
Computational Approach to Riemann Surfaces by Alexander I. Bobenko (auth.), Alexander I. Bobenko, Christian Klein (eds.)