Riemann surface

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1: 21.7 Riemann Surfaces

§21.7 Riemann Surfaces

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§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces

… ►Removing the singularities of this curve gives rise to a two-dimensional connected manifold with a complex-analytic structure, that is, a Riemann surface. All compact Riemann surfaces can be obtained this way. … ►

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Figure 21.7.1: A basis of cycles for a genus 2 surface. Magnify

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§21.7(iii) Frobenius’ Identity

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2: 21.10 Methods of Computation

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§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface

►In addition to evaluating the Fourier series, the main problem here is to compute a Riemann matrix originating from a Riemann surface. … ►

Belokolos et al. (1994, Chapter 5) and references therein. Here the Riemann surface is represented by the action of a Schottky group on a region of the complex plane. The same representation is used in Gianni et al. (1998).

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Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.

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Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

3: Bernard Deconinck

… ►He has worked on integrable systems, algorithms for computations with Riemann surfaces, Bose-Einstein condensates, and methods to investigate the stability of solutions of nonlinear wave equations. He is the coauthor of several Maple commands to work with Riemann surfaces and the command to compute multidimensional theta functions numerically. …

4: 21.9 Integrable Equations

… ►Particularly important for the use of Riemann theta functions is the Kadomtsev–Petviashvili (KP) equation, which describes the propagation of two-dimensional, long-wave length surface waves in shallow water (Ablowitz and Segur (1981, Chapter 4)): …These parameters, including $𝛀$, are not free: they are determined by a compact, connected Riemann surface (Krichever (1976)), or alternatively by an appropriate initial condition $u(x,y,0)$ (Deconinck and Segur (1998)). … ►

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Figure 21.9.2: Contour plot of a two-phase solution of Equation (21.9.3). … Magnify

►Furthermore, the solutions of the KP equation solve the Schottky problem: this is the question concerning conditions that a Riemann matrix needs to satisfy in order to be associated with a Riemann surface (Schottky (1903)). Following the work of Krichever (1976), Novikov conjectured that the Riemann theta function in (21.9.4) gives rise to a solution of the KP equation (21.9.3) if, and only if, the theta function originates from a Riemann surface; see Dubrovin (1981, §IV.4). …

5: 21.1 Special Notation

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$g,h$	positive integers.
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$a\circ b$	intersection index of $a$ and $b$, two cycles lying on a closed surface. $a\circ b=0$ if $a$ and $b$ do not intersect. Otherwise $a\circ b$ gets an additive contribution from every intersection point. This contribution is $1$ if the basis of the tangent vectors of the $a$ and $b$ cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is $-1$.
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6: 21.4 Graphics

… ►Figure 21.4.1 provides surfaces of the scaled Riemann theta function $\widehat{\theta }\left(𝐳\right|𝛀)$, with …This Riemann matrix originates from the Riemann surface represented by the algebraic curve ${\mu }^3-{\lambda }^7+2{\lambda }^3\mu =0$; compare §21.7(i). ►

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Figure 21.4.1: $\widehat{\theta }\left(𝐳\right|𝛀)$ parametrized by (21.4.1). The surface plots are of $\widehat{\theta }\left(x+\mathrm{i}y,0\right|𝛀)$, $0\le x\le 1$, $0\le y\le 5$ (suffix 1); $\widehat{\theta }\left(x,y\right|𝛀)$, $0\le x\le 1$, $0\le y\le 1$ (suffix 2); $\widehat{\theta }\left(\mathrm{i}x,\mathrm{i}y\right|𝛀)$, $0\le x\le 5$, $0\le y\le 5$ (suffix 3). …

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Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: $\mathrm{\Re }\widehat{\theta }\left(x+\mathrm{i}y,0,0\right|{𝛀}_2)$, $0\le x\le 1$, $0\le y\le 3$. This Riemann matrix originates from the genus 3 Riemann surface represented by the algebraic curve ${\mu }^3+2\mu -{\lambda }^4=0$; compare §21.7(i). Magnify 3D Help

7: Bibliography R

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Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.

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External Links:

MathReview (H. H. Martens), ZentralBlatt

Referenced by:

§20.11(iv), §20.12(ii).

Encodings:

BibTeX

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8: Bibliography T

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Combinatorial Group Theory, Riemann Surfaces and Differential Equations. In Contributions to Group Theory, Contemp. Math., Vol. 33, pp. 467–519.

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External Links:

MathReview (William Harvey), ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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9: Bibliography F

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Theta Functions on Riemann Surfaces. Springer-Verlag, Berlin.

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Notes:

Lecture Notes in Mathematics, Vol. 352

External Links:

MathReview (H. M. Farkas), ZentralBlatt

Referenced by:

§21.1, §21.7(ii), Ch.21.

Encodings:

BibTeX

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10: Bibliography J

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Java Tools for Experimental Mathematics

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Notes:

Single and double precision. Contains packages to compute elliptic functions, Riemann theta functions and their derivatives, and Riemann matrices for Riemann surfaces.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

Encodings:

BibTeX

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