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We return to the conservation law (1.2), but now study the case of systems, i.e., $$\displaystyle u_{t}+f(u)_{x}=0,$$ (5.1) where \(u=u(x,t)=(u_{1},\dots,u_{n})\) and \(f=f(u)=(f_{1},\dots,f_{n})\in C^{2}\) are vectors in \(\mathbb{R}^{n}\). (We will not distinguish between row and column vectors, and use whatever is more convenient.) Furthermore, in this chapter we will consider only systems on the line; i.e., the dimension of the underlying physical space is still one. In Chapt. 2 we proved existence, uniqueness, and stability of the Cauchy problem for the scalar conservation law in one space dimension, i.e., well-posedness in the sense of Hadamard. However, this is a more subtle question in the case of systems of hyperbolic conservation laws. We will here first discuss the basic concepts for systems: fundamental properties of shock waves and rarefaction waves. In particular, we will discuss various entropy conditions to select the right solutions of the Rankine–Hugoniot relations.

As we have seen in Chap. 1, a Euclidean triangulated surface (T l , M) characterizes a polyhedral metric with conical singularities associated with the vertices of the triangulation. In this chapter we show that around any such a vertex we can introduce complex coordinates in terms of which we can write down the conformal conical metric, locally parametrizing the singular structure of (T l , M). This makes available a powerful dictionary between 2-dimensional triangulations and complex geometry. It must be noted that, both in the mathematical and in the physical applications of the theory, the connection between Riemann surfaces and triangulations typically emphasizes only the role of ribbon graphs and of the associated metric. The conical geometry of the polyhedral surface (T l , M) is left aside and seems to play no significant a role. Whereas this attitude can be motivated by the observation (due to M. Troyanov, On the moduli space of singular Euclidean surfaces. In: Papadopoulos A (ed) Handbook of teichmuller theory, vol I. IRMA lectures in mathematics and theoretical physics, vol 11. European Mathematical Society (EMS), Zurich, pp 507–540, arXiv:math/0702666v2 [math.DG], 2007) that the conformal structure does not see the conical singularities (see below, for details), it gives a narrow perspective of the much wider role that the theory has to offer. Thus, it is more appropriate to connect a polyhedral surface (T l , M) to a corresponding Riemann surface by taking fully into account the conical geometry of (T l , M). This connection is many–faceted and exploits a vast repertoire of notion ranging from complex function theory to algebraic geometry. We start by defining the barycentrically dual polytope (P T , M) associated with a polyhedral surface (T l , M) and discuss the geometry of the corresponding ribbon graph. Then, by adapting to our case the elegant approach in Mulase and Penkava (Asian J Math 2(4):875–920, 1998. math-ph/9811024 v2), we explicitly construct the Riemann surface associated with (P T , M). This directly bring us to the analysis of Troyanov’s singular Euclidean structures and to the construction of the bijective map between the moduli space \(\mathfrak {M}_{g,N_0}\) of Riemann surfaces (M, N0) with N0 marked points, decorated with conical angles, and the space of polyhedral structures. In particular the first Chern class of the line bundles naturally defined over \(\mathfrak {M}_{g,N_0}\) by the cotangent space at the i-th marked point is related with the corresponding Euler class of the circle bundles over the space of polyhedral surfaces defined by the conical cotangent spaces at the i-th vertex of (T l , M). Whereas this is not an unexpected connection, the analogy with Witten–Kontsevich theory being obvious, we stress that the conical geometry adds to this property the possibility of a deep and explicit characterization of the Weil–Petersson form in terms of the edge–lengths of (T l , M). This result (Carfora et al., JHEP 0612:017, 2006. arXiv:hep-th/0607146) is obtained by a subtle interplay between the geometry of (T l , M) and 3-dimensional hyperbolic geometry, and it will be discussed in detail in Chap. 3 since it explicitly hints to the connection between polyhedral surfaces and quantum geometry in higher dimensions.

In contrast to a domain D ⊂ ℂ, where rational functions already provide a big class of meromorphic functions, on a Riemann surface it is not possible without further effort to construct meromorphic functions, for example with a given finite set of poles. Constructive problems of this kind are central problems in the theory of Riemann surfaces. It turns out that it is easier to construct harmonic instead of analytic functions. So we are led to pick up a thread which we dropped very early on in the first volume [FB] (Chap. I, Sect. 5). Our treatment is very much based on that in the classic book by Nevanlinna [Ne].

Introduction to Compact Riemann Surfaces.- Computing with plane algebraic curves and Riemann surfaces: the algorithms of the Maple package "algcurves".- Algebraic curves and Riemann surfaces in Matlab.- Computing Poincare Theta Series for Schottky Groups.- Uniformizing real hyperelliptic M-curves using the Schottky-Klein prime function.- Numerical Schottky Uniformizations: Myrberg's Opening Process.- Period Matrices of Polyhedral Surfaces.- On the spectral theory of the Laplacian on compact polyhedral surfaces of arbitrary genus.

In this chapter, we present an overview of different algorithms for computing with compact connected Riemann surfaces, obtained from desingularized and compactified plane algebraic curves. As mentioned in Chap.1 [Bob11], all compact connected Riemann surfaces may be represented this way. The Maple package “algcurvess”, largely developed by the authors and Mark van Hoeij contains implementations of these algorithms. A few recent additions to the “algcurves” package are not due to the authors or Mark van Hoeij. The algorithm behind those commands are not discussed here as they have no bearing on anything associated with Riemann surfaces.

In the previous chapter, a detailed description of the algorithms for the ‘algcurves’ package in Maple was presented. As discussed there, the package is able to handle general algebraic curves with coefficients given as exact arithmetic expressions, a restriction due to the use of exact integer arithmetic. Coefficients in terms of floating point numbers, i.e., the representation of decimal numbers of finite length on a computer, can in principle be handled, but the floating point numbers have to be converted to rational numbers. This can lead to technical difficulties in practice. One also faces limitations if one wants to study families of Riemann surfaces, where the coefficients in the algebraic equation defining the curve are floating point numbers depending on a set of parameters, i.e., if one wants to explore modular properties of Riemann surfaces as in the examples discussed below. An additional problem in this context can be computing time since the computation of the Riemann matrix uses the somewhat slow Maple integration routing. Thus, a more efficient computation of the Riemann matrix is interesting if one wants to study families of Riemann surfaces or higher genus examples which are computationally expensive.

As said in the Introduction, under the correspondence between compact Riemann surfaces and smooth irreducible complex projective algebraic curves, the fact that a Riemann surface S is symmetric means that the corresponding complex curve C can be defined over the field R of real numbers.

Preliminaries.- On the Number of Conjugacy Classes of Symmetries of Riemann Surfaces.- Counting Ovals of Symmetries of Riemann Surfaces.- Symmetry Types of Some Families of Riemann Surfaces.- Symmetry Types of Riemann Surfaces with a Large Group of Automorphisms.

Throughout this chapter we will be involved in several enumerations of ovals of the symmetries of a Riemann surface S. Recall that by an oval of a symmetry s we mean a connected component of the fixed point set Fix(s) of s. The first section is short but crucial for the rest of the monograph.

In this chapter we study the symmetries of those surfaces S whose group Aut+(S) of analytic automorphisms is large. As said in the introduction, the symmetries of these surfaces are topologically determined by Aut+(S).

Riemann surfaces naturally appear in the analysis of complex functions that are branched over the complex plane. However, they usually possess a complicated topology and are thus hard to understand. We present an algorithm for constructing Riemann surfaces as meshes in ${\mathbb R}^3$ from explicitly given branch points with corresponding branch indices. The constructed surfaces cover the complex plane by the canonical projection onto ${\mathbb R}^2$ and can therefore be considered as multivalued graphs over the plane – hence they provide a comprehensible visualization of the topological structure. Complex functions are elegantly visualized using domain coloring on a subset of ${\mathbb C}$. By applying domain coloring to the automatically constructed Riemann surface models, we generalize this approach to deal with functions which cannot be entirely visualized in the complex plane.

In this paper we introduce computable and time bounded Riemann surfaces, based on the classical abstract definition by charts. Building upon this definition we discuss computable versions of several classical results, such as the existence of complete continuations of holomorphic functions, universal coverings and the uniformization theorem (for some cases). Though we state most of our results for computable surfaces, many of them can also be transformed to a uniform version, i.e. based on representations of the class of Riemann surfaces (modulo conformal equivalence).

Let M be a differentiable manifold of dimension d. We would first like to recall some basic properties of differential forms on such a manifold.

We study an explicit construction of C*-algebraic deformation quantization of closed Riemann surfaces.

This book is novel in its broad perspective on Riemann surfaces: the text systematically explores the connection with other fields of mathematics. The book can serve as an introduction to contemporary mathematics as a whole, as it develops background material from algebraic topology, differential geometry, the calculus of variations, elliptic PDE, and algebraic geometry. The book is unique among textbooks on Riemann surfaces in its inclusion of an introduction to Teichmuller theory. For this new edition, the author has expanded and rewritten several sections to include additional material and to improve the presentation.

The present paper is a detailed exposition of the author’s report delivered at the International Sympozium in Memory of Hua Loo Keng, Beijing, on August the 6th, 1988.

1. The Jacobian varieties provide an effective and natural tool for the study of certain problems in the theory of closed Riemann surfaces. Their classical construction is very simple: 1 Let X be a closed Riema.nn surface of genus g ~ i. Let ~ ,...,~g be a basis for the holomorphic differentials, and let yi,72,...,72g be a basis for the first integral homology group, on X. The associated period matrix Z : (lj) is defined by i Ii ~ j w ~j I t s c o l u m n s a r e N l i n e a r l y i n d e p e n d e n t a n d g e n e r a t e a g r o u p o f t r a n s l a t i o n s F o f C g . T h e c o m p l e x t o r u s J ( X ) = c g / F w i l l b e r e f e r r e d t o a s a J a e o b i a n v a r i e t y o f X. Let P be a fixed reference point of X. For an arbitrary point Q { X the point u(Q) 6 C g with coordinates Q i r i u (Q) =J

This note contains a proof of an elementary but useful inequality for Riemann surfaces with a finitely generated non-Abelian fundamental group.

We restrict ourselves at first to the class considered in XI, § 2, of simply connected Riemann surfaces F q whose branch points project onto a finite number q of image points w 1, .... w q . These surfaces can be represented concretely by the line complexes defined in the place just mentioned. To each half sheet of F q there corresponds a node point in the line complex. By means of the line segments connecting the node points, the plane is decomposed into elementary regions which correspond one-to-one to the branch points of F in such a way that one elementary region with 2m sides is associated with a branch point of order m — 1. Corresponding in the complex to the sheets which are unramified over the base points are two-sided figures (double 1ine segments).

Let W be an open Riemann surface. We say that W satisfies the absolute A B-maximum principle if every subregion V of W with compact relative boundary Г has the property that each function which is bounded and analytic on V ∪ Г assumes its maximum on Г. Henceforth, we shall restrict ourselves to subregions whose relative boundaries consist of a finite number of piecewise analytic curves.

In 1956 Z. Kuramochi [2] defined a new boundary for any open Riemann surface R. It is now called Kuramochi boundary. Rigorous treatments of this boundary are found in Ill and [4]. It shares some properties with inner points of R. For instance, the values of every SHS function 2) are defined on the Kuramochl boundary in [1]. This was also done by Kuramochl [3] but his discussions were not quite clear.