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Ashley Tucker
Ashley Tucker

Riemann Surfaces

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

Riemann Surfaces

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The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm.

The existence of non-constant meromorphic functions can be used to show that any compact Riemann surface is a projective variety, i.e. can be given by polynomial equations inside a projective space. Actually, it can be shown that every compact Riemann surface can be embedded into complex projective 3-space. This is a surprising theorem: Riemann surfaces are given by locally patching charts. If one global condition, namely compactness, is added, the surface is necessarily algebraic. This feature of Riemann surfaces allows one to study them with either the means of analytic or algebraic geometry. The corresponding statement for higher-dimensional objects is false, i.e. there are compact complex 2-manifolds which are not algebraic. On the other hand, every projective complex manifold is necessarily algebraic, see Chow's theorem.

Topologically there are only three types: the plane, the cylinder and the torus. But while in the two former case the (parabolic) Riemann surface structure is unique, varying the parameter τ \displaystyle \tau in the third case gives non-isomorphic Riemann surfaces. The description by the parameter τ \displaystyle \tau gives the Teichmüller space of "marked" Riemann surfaces (in addition to the Riemann surface structure one adds the topological data of a "marking", which can be seen as a fixed homeomorphism to the torus). To obtain the analytic moduli space (forgetting the marking) one takes the quotient of Teichmüller space by the mapping class group. In this case it is the modular curve.

For example, hyperbolic Riemann surfaces are ramified covering spaces of the sphere (they have non-constant meromorphic functions), but the sphere does not cover or otherwise map to higher genus surfaces, except as a constant.

The classification scheme above is typically used by geometers. There is a different classification for Riemann surfaces which is typically used by complex analysts. It employs a different definition for "parabolic" and "hyperbolic". In this alternative classification scheme, a Riemann surface is called parabolic if there are no non-constant negative subharmonic functions on the surface and is otherwise called hyperbolic.[4][5] This class of hyperbolic surfaces is further subdivided into subclasses according to whether function spaces other than the negative subharmonic functions are degenerate, e.g. Riemann surfaces on which all bounded holomorphic functions are constant, or on which all bounded harmonic functions are constant, or on which all positive harmonic functions are constant, etc.

A Riemann surface is a surface-like configuration that covers the complex plane with several, and in general infinitely many, "sheets." These sheets can have very complicated structures and interconnections (Knopp 1996, pp. 98-99). Riemann surfaces are one way of representing multiple-valued functions; another is branch cuts. The above plot shows Riemann surfaces for solutions of the equation

The Riemann surface of the function field is the set of nontrivial discrete valuations on . Here, the set corresponds to the ideals of the ring of integers of over . ( consists of the elements of that are roots of monic polynomials over .) Riemann surfaces provide a geometric visualization of functions elements and their analytic continuations.

I know that all compact Riemann surfaces with the same genus are topologically equivalent. Moreover they are diffeomorphic. But are they biholomorphic, too?In other words, is the complex structure conserved?

Some magic words for this question are "moduli space" or "moduli stack". In the early days, one was interested in a variety or variety-like object which would classify projective complex curves (compact Riemann surfaces) of given genus $g$, i.e., whose points correspond to isomorphism classes of curves (or biholomorphism classes of compact Riemann surfaces). This is nowadays called a "coarse moduli space". As GH and François commented, there is a whole continuum of points in the coarse moduli space of genus 1; the same is true for any genus $g > 1$.

The answer is no. For example, if $\Lambda_1$ and $\Lambda_2$ are two lattices in $\mathbbC$, then the surfaces $\mathbbC/\Lambda_1$ and $\mathbbC/\Lambda_2$ are conformally equivalent if and only if $\Lambda_1$ and $\Lambda_2$ are similar. This follows from the theory of elliptic functions (or elliptic curves).

In three-dimensional (3D) graphene structures, some interesting topology like helical spirals from graphite screw dislocations has been proposed25. Moreover, four kinds of dislocations and helical shapes were observed in raw anthracite using the bright-field high-resolution transmission electron microscopy (HRTEM)26. For the present case of nanosolenoids, one atomic graphene plane continuously spirals around the line perpendicular to the basal plane, which can be considered to closely follow a Riemann surface (namely, a log z type). As well-known objects in mathematics, Riemann surfaces (Fig. 1c and Supplementary Fig. 1, representative examples of Riemann surfaces) were proposed by Riemann in 1851 to predict a single-valued domain for a multivalued analytical function. It is noteworthy that the Riemann surfaces not only play key roles in the development of modern mathematics but also provide insights for the design and synthesis of multifunctional curved carbon materials25,26,27,28. In 2016, Yakobson and coworkers initially predicted that a carbon solenoid with Riemann surfaces and small diameter can behave as a quantum conductor when a voltage is applied, resulting in a large magnetic field near the center and bringing about excellent inductance25.

EDIT: After what has developed, I feel this question is now appropriate: Is there a complex analysis text that would be particularly recommended if one wishes to study Riemann surfaces? What topics in particular is it important to develop a good grasp of?

It is completely geometric and will introduce you, starting from scratch, not only to Riemann surfaces but also to the theory or holomorphic functions of several variables, covering spaces, cohomology,...This unique book emphasizes how little you have to know of the classical function of one complex variable: just the forty pages of Chapter 1, aptly named Elementary Theory of Holomorphic Functions. A book with a similar philosophy is Analyse Complexe by Dolbeault, he of the Dolbeault cohomology, which has the drawback of being in French (albeit in mathematical French, which is a far cry from Mallarmé or Proust French...)

It is an underappreciated fact, displayed in both these books, that most of the material found in books on complex analysis of one variable is useless for the study of Riemann surfaces and more generally complex manifolds.For example all the clever computations of real integrals by residue calculus, evaluation of convergence radius of power series, asymptotic methods, Weierstraß products, Schwarz-Christoffel transformations, ... are irrelevant in complex analytic geometry: I challenge anyone to find the slightest trace of these in the work of the recently deceased H. Grauert, arguably the greatest 20th century specialist in the geometry of complex analytic spaces.

We can also work with Riemann surfaces that are defined over fields with acomplex embedding, but since the current interface for computing genus andregular differentials in Singular presently does not support extensions ofQQ, we need to specify a description of the differentials ourselves. We givean example of a CM elliptic curve:

Let \(\left(I M \right)\) be the normalized period matrix (\(M\) is the\(g\times g\) riemann_matrix()). We consider the system of matrixequations \(MA + C = (MB + D)M\) where \(A, B, C, D\) are \(g\times g\)integer matrices. We determine small integer (near) solutions using LLLreductions. These solutions are returned as \(2g \times 2g\) integermatrices obtained by stacking \(\left(D B\right)\) on top of \(\left(C A\right)\).

Rudimentary class to represent disjoint unions of Riemann surfaces. Existsmainly (and this is the only functionality actually implemented) torepresents direct products of the complex tori that arise as analyticJacobians of Riemann surfaces.

Definition 1 (Riemann surface) If is a Hausdorff connected topological space, a (one-dimensional complex) atlas is a collection of homeomorphisms from open subsets of that cover to open subsets of the complex numbers , such that the transition maps defined by are all holomorphic. Here is an arbitrary index set. Two atlases , on are said to be equivalent if their union is also an atlas, thus the transition maps and their inverses are all holomorphic. A Riemann surface is a Hausdorff connected topological space equipped with an equivalence class of one-dimensional complex atlases.A map from one Riemann surface to another is holomorphic if the maps are holomorphic for any charts , of an atlas of and respectively; it is not hard to see that this definition does not depend on the choice of atlas. It is also clear that the composition of two holomorphic maps is holomorphic (and in fact the class of Riemann surfaces with their holomorphic maps forms a category). 041b061a72


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