Automorphisms of compact Riemann surfaces

Consider a compact Riemann surface (or smooth projective algebraic curve in characteristic zero) . One of the first facts one observes in their theory is that the group of automorphisms of is quite large when has genus zero or one. When has genus zero, it is the projective line, and its automorphism group is , a fact which generalizes naturally to higher projective spaces. When , the curve acquires the structure of an elliptic curve from any distinguished point. Thus, translations by any element of act on . So has points corresponding to each element of (and a few more, such as inversion in the group law).

But when the genus of is at least two, things change dramatically. It is a famous theorem that the number of automorphisms is bounded:

Theorem 1 (Hurwitz) If is a compact Riemann surface of genus , then there are at most automorphisms of .

I won’t write out a proof here; a discussion is in lecture 9 of the notes I’m taking in an algebraic curves class. In fact, the hard part is to show that there are finitely many automorphisms, after which it is a combinatorial argument.

This bound is often sharp. There are infinitely many genera together with Riemann surfaces with exactly automorphisms; there are explicit constructions that I’m not very familiar with. It is false in characteristic , because the Riemann-Hurwitz formula is no longer necessarily true (because of the existence of non-separable morphisms), and counterexamples are given in the notes.

What I want to describe today is that this bound is often not sharp.

Theorem 2 If is a compact Riemann surface of genus for a prime, then has order strictly less than .

 

Existence theorems for harmonic and meromorphic functions

Now we’re going to use the machinery already developed to prove the existence of harmonic functions.

Fix a Riemann surface and a coordinate neighborhood isomorphic to the unit disk in (in fact, I will abuse notation and identify the two for simplicity), with corresponding to .

First, one starts with a function such that:

1. is the restriction of a harmonic function on some 2. on the boundary (this is a slight abuse of notation, but ok in view of 1).

The basic example is .

Theorem 1 There is a harmonic function such that is continuous at , and if is a bounded smooth function that vanishes in a neighborhood of .

 

In other words, we are going to get harmonic functions that are not globally defined, but whose singularities are localized.

 

Smoothness properties of the decomposition of L^2(M)

I realize I’ve been slow as of late.

Though the decomposition of the square-integrable 1-forms was a Hilbert space, -decomposition, it reflects many smoothness properties in which we are interested.  The goal of this post is twofold: first, show that the smooth differentials in the first two components are respectively the exact and coexact ones; and second, show that the closedness of a square-integrable 1-form in some neighborhood  implies the smoothness of the corresponding term.  The ultimate goal is the existence theorems for harmonic functions on Riemann surfaces.

Smooth differentials in

I now claim that any smooth differential is exact and any smooth differential in is co-exact. This is nontrivial because of the way we took completions. I will only prove the claim for .

We already have closedness of by the previous post. To show exactness, it will be sufficient to show that for any closed smooth curve ,

because we could use a path integral to define the antiderivative. We can approximate by a simple closed curve homotopic to , so we can assume at the outset that is a homeomorphism . 

Proposition 1 Given as above, there is a closed differential such that  

for any closed . 

To prove this, find an annular neighborhood of the compact set , which is divided into as in the figure.

 

Decomposition of the space of square-integrable 1-forms

Yesterday I defined the Hilbert space of square-integrable 1-forms on a Riemann surface . Today I will discuss the decomposition of it. Here are the three components:

1) is the closure of 1-forms where is a smooth function with compact support.

2) is the closure of 1-forms where is a smooth function with compact support.

3) is the space of square-integrable harmonic forms.

Today’s goal is:

Theorem 1 As Hilbert spaces,

  

The proof will be divided into several steps.

 

More about 1-forms on Riemann surfaces

It’s now time to do some more manipulations with differential forms on a Riemann surface. This will establish several notions we will need in the future.

The Hodge star

Given the 1-form in local coordinates as , define

In other words, given the decomposition , we act by on the first sumamand and by on the second. This shows that the operation is well-defined. Note that is conjugate-linear and . Also, we see that if .  This operation is called the Hodge star.

From the latter description of the Hodge star we see that for any smooth ,

From the definitions of , this can be written as if is the usual Laplacian with respect to the local coordinates .

The Hodge star allows us to define co things. A form is co-closed if ; it is co-exact if for smooth.

 

Riemann surfaces

Finally, it’s time to get to the definition of a Riemann surface. A Riemann surface is a connected one-dimensional complex manifold. In other words, it’s a Hausdorff space which is locally homeomorphic to via charts (i.e., homeomorphisms) for open and such that is holomorphic.

Examples

Evidently an open subset of a Riemann surface is a Riemann surface. In particular, an open subset of is a Riemann surface in a natural manner.

The Riemann sphere or is a Riemann sphere with the open sets and the charts

The transition map is and thus holomorphic on .

An important example comes from analytic continuation, which I will briefly sketch below. A function element is a pair where is holomorphic and is an open disk. Two function elements are said to be direct analytic continuations of each other if and on . By piecing together direct analytic continuations on a curve, we can talk about the analytic continuation of a function element along a curve (which may or may not exist, but if it does, it is unique).

Starting with a given function element , we can consider the totality of all equivalence classes of function elements that can be obtained by continuing along curves in . Then is actually a Riemann surface.