Abbas - An Introduction to Compactness Results in Symplectic Field Theory

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Casim Abbas An Introduction to Compactness Results in Symplectic Field Theory 2014 10.1007/978-3-642-31543-5_1

Springer-Verlag Berlin Heidelberg 2014

1. Riemann Surfaces

Casim Abbas 1

(1)

Department of Mathematics, Michigan State University, East Lansing, MI, USA

Abstract

In this chapter we define the notion of convergence of a sequence of Riemann surfaces, and we prove that any sequence of smooth stable Riemann surfaces has a subsequence which converges to a noded Riemann surface. This is a special case of the celebrated result by P. Deligne and D. Mumford concerning the compactification of the moduli space of algebraic curves. We follow an approach by W. Thurston which is more geometric in nature, viewing Riemann surfaces as surfaces equipped with a hyperbolic metric. The exposition has been made self-contained because the details are scattered throughout the existing literature. In particular, we explain all the necessary background material from hyperbolic geometry.

The purpose of this chapter is to provide the material necessary for understanding convergence of Riemann surfaces in the sense of DeligneMumford. If we want to discuss the convergence behavior of a sequence of J -holomorphic curves ( u k ) we need to take into account that their domains are all different Riemann surfaces ( S , j k ). The aim is to establish a suitable notion of convergence for these as well. The discussion here follows Thurstons approach [] as general references). The details, however, are spread out over the literature so we give a thorough and unified presentation.

1.1 Smooth and Noded Riemann Surfaces

Definition 1.1

An atlas on a smooth surface S with charts , U = S is called conformal if the transition maps

are holomorphic. A conformal structure on S is a maximal conformal atlas. A smooth surface S together with a conformal structure is called a Riemann surface . A continuous map f : S S between two Riemann surfaces is called holomorphic if in local coordinates { U , } on S and on S the maps are holomorphic whenever they are defined. A holomorphic map f : S S is called conformal if its derivative is never zero.

It is common practice to identify a coordinate patch U S with its image suppressing the coordinate map in the notation. This makes sense if we consider local objects and notions which are invariant under conformal maps. For example, if is a smooth function on a Riemann surface then we may define f is subharmonic by the requirement that . This is well-defined if for any locally defined conformal map z = h ( w ) the map f h satisfies . But this follows from

From this it is clear that the local expression for example would not make sense globally on a Riemann surface. The following statements hold (the proofs are trivial and we leave them to the reader):

(1)

Every Riemann surface is orientable.

(2)

Assume S is a Riemann surface, S is a smooth surface and : S S is a local diffeomorphism. Then there exists a unique conformal structure on S such that becomes holomorphic.

(3)

Application: Every covering space of a Riemann surface can be made into a Riemann surface in a canonical fashion.

(4)

Let S be a Riemann surface, let S be a smooth surface and let : S S be a covering. If every covering transformation : S S is holomorphic then there is a unique conformal structure on S such that becomes holomorphic.

There are different ways to look at Riemann surfaces. We will elaborate on some of them and explain why they are equivalent to Definition 1.1.

Definition 1.2

(Almost complex structure-Alternative definition of Riemann surface)

Let W be a differentiable manifold. An almost complex structure on M is a smooth section j of the vector bundle , such that for all z M . A pair ( S , j ) consisting of a smooth surface S and an almost complex structure j is called a Riemann surface .

The two notions of Riemann surface are equivalent: Assume S is a smooth surface together with a conformal structure. Then we can define an almost complex structure j on S as follows: Let z S . Let be a coordinate chart belonging to the conformal structure. We define

(1.1)

Show as an exercise that this definition does not depend on the choice of the coordinate chart . This procedure also works for general complex manifolds, i.e. smooth manifolds of even dimension which admit an atlas with holomorphic coordinate transition maps. The converse procedure would be constructing a conformal structure on a manifold with a given almost complex structure j such that () holds.

Definition 1.3

Assume that W is a manifold with an almost complex structure j . If W admits an atlas such that the transition maps between coordinate patches are all holomorphic, and if j is then given by () then it is called a complex structure or an integrable almost complex structure .

Remark 1.4

The existence question of an almost complex structure on a given even dimensional manifold M is rather a topological question while the existence question of an integrable almost complex structure is of analytical nature, and it is usually more difficult to answer. For example, it is known that the only spheres admitting almost complex structures are S 2 and S 6. By our remarks above every almost complex structure on S 2 is integrable. On the other hand, it is an open question whether S 6 admits any complex structure.

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