Consider a compact Riemann surface (or smooth projective algebraic curve in characteristic zero) ${X}$. One of the first facts one observes in their theory is that the group ${\mathrm{Aut}(X)}$ of automorphisms of ${X}$ is quite large when ${X}$ has genus zero or one. When ${X}$ has genus zero, it is the projective line, and its automorphism group is ${\mathrm{PGL}_2(\mathbb{C})}$, a fact which generalizes naturally to higher projective spaces. When ${g = 1}$, the curve ${X}$ acquires the structure of an elliptic curve from any distinguished point. Thus, translations by any element of ${X}$ act on ${X}$. So ${\mathrm{Aut}(X)}$ has points corresponding to each element of ${X}$ (and a few more, such as inversion in the group law).

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

Theorem 1 (Hurwitz) If ${X}$ is a compact Riemann surface of genus ${g \geq 2}$, then there are at most ${84(g-1)}$ automorphisms of ${X}$.

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 ${g}$ together with Riemann surfaces with exactly ${84(g-1)}$ automorphisms; there are explicit constructions that I’m not very familiar with. It is false in characteristic $p$, 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 ${X}$ is a compact Riemann surface of genus ${g = p+1}$ for ${p>84}$ a prime, then ${\mathrm{Aut}(X)}$ has order strictly less than ${84(g-1)}$. (more…)

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

Fix a Riemann surface ${M}$ and a coordinate neighborhood ${(U,z)}$ isomorphic to the unit disk ${D_1}$ in ${\mathbb{C}}$ (in fact, I will abuse notation and identify the two for simplicity), with ${P \in M}$ corresponding to ${0}$.

First, one starts with a function ${h: D_1 - \{0 \} \rightarrow \mathbb{C}}$ such that:

1. ${h}$ is the restriction of a harmonic function on some ${D_{1+\epsilon} - 0}$ 2. ${d {}^* h = 0}$ on the boundary ${\partial D_1}$ (this is a slight abuse of notation, but ok in view of 1).

The basic example is ${z^{-n} + \bar{z}^{n}}$.

Theorem 1 There is a harmonic function ${f: M - P \rightarrow \mathbb{C}}$ such that ${f-h}$ is continuous at ${P}$, and ${\phi df \in L^2(M)}$ if ${\phi}$ is a bounded smooth function that vanishes in a neighborhood of ${P}$.

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

I realize I’ve been slow as of late.

Though the decomposition of the square-integrable 1-forms was a Hilbert space, $L^2$-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 $E$ term.  The ultimate goal is the existence theorems for harmonic functions on Riemann surfaces.

Smooth differentials in ${E, E^*}$

I now claim that any smooth differential ${\omega \in E}$ is exact and any smooth differential in ${E^*}$ is co-exact. This is nontrivial because of the way we took completions. I will only prove the claim for ${E}$.

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

$\displaystyle \int_c \omega = 0$

because we could use a path integral to define the antiderivative. We can approximate ${c}$ by a simple closed curve homotopic to ${c}$, so we can assume at the outset that ${c}$ is a homeomorphism ${S^1 \rightarrow M}$

Proposition 1 Given ${c}$ as above, there is a closed differential ${\eta_c}$ such that$\displaystyle ( \omega, {}^* \eta_c) = \int_c \omega$

for any closed ${\omega}$

To prove this, find an annular neighborhood of the compact set ${c(S^1) \subset M}$, which is divided into ${R_1, R_2}$ as in the figure.

Yesterday I defined the Hilbert space of square-integrable 1-forms ${L^2(X)}$ on a Riemann surface ${X}$. Today I will discuss the decomposition of it. Here are the three components:

1) ${E}$ is the closure of 1-forms ${df}$ where ${f}$ is a smooth function with compact support.

2) ${E^*}$ is the closure of 1-forms ${{}^* df}$ where ${f}$ is a smooth function with compact support.

3) ${H}$ is the space of square-integrable harmonic forms.

Today’s goal is:

Theorem 1 As Hilbert spaces,

$\displaystyle L^2(X) = E \oplus E^* \oplus H.$

The proof will be divided into several steps. (more…)

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 ${\omega}$ in local coordinates as ${u dz + v d\bar{z}}$, define

$\displaystyle ^*{\omega} := -iu dz + iv d\bar{z} .$

In other words, given the decomposition ${T^*(X) = T^{*(1,0)}(X) \oplus T^{*(0,1)}(X)}$, we act by ${-i}$ on the first sumamand and by ${i}$ on the second. This shows that the operation is well-defined. Note that ${^*{}}$ is conjugate-linear and ${^*{}^2 = -1}$. Also, we see that ${^*{} dx = dy, ^*{dy} = -dx}$ if ${z = x + iy}$.  This operation is called the Hodge star.

From the latter description of the Hodge star we see that for any smooth ${f}$,

$\displaystyle d ^*{} df = d( -if_z dz + if_{\bar{z}} d\bar{z}) = 2i f_{z \bar{z}} dz \wedge d\bar{z}.$

From the definitions of ${f_{z}, f_{\bar{z}}}$, this can be written as ${-2i \Delta f dz \wedge d\bar{z}}$ if ${\Delta}$ is the usual Laplacian with respect to the local coordinates ${x,y}$.

The Hodge star allows us to define co things. A form ${\omega}$ is co-closed if ${d ^*{} \omega = 0}$; it is co-exact if ${\omega = ^* df}$ for ${f}$ smooth. (more…)

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 ${M}$ which is locally homeomorphic to ${\mathbb{C}}$ via charts (i.e., homeomorphisms) ${\phi_i:U_i \rightarrow V_i}$ for ${U_i \subset M, V_i \subset \mathbb{C}}$ open and such that ${\phi_j \circ \phi_i^{-1}: V_i \cap V_j \rightarrow V_i \cap V_j}$ is holomorphic.

Examples

Evidently an open subset of a Riemann surface is a Riemann surface. In particular, an open subset of ${\mathbb{C}}$ is a Riemann surface in a natural manner.

The Riemann sphere ${P^1(\mathbb{C}) := \mathbb{C} \cup \{ \infty \}}$ or ${S^2}$ is a Riemann sphere with the open sets ${U_1 = \mathbb{C}, U_2 = \mathbb{C} - \{0\} \cup \{\infty\}}$ and the charts

$\displaystyle \phi_1 =z, \ \phi_2 = \frac{1}{z}.$

The transition map is ${\frac{1}{z}}$ and thus holomorphic on ${U_1 \cap U_2 = \mathbb{C}^*}$.

An important example comes from analytic continuation, which I will briefly sketch below. A function element is a pair ${(f,V)}$ where ${f: V \rightarrow \mathbb{C}}$ is holomorphic and ${V \subset \mathbb{C}}$ is an open disk. Two function elements ${(f,V), (g,W)}$ are said to be direct analytic continuations of each other if ${V \cap W \neq \emptyset}$ and ${f \equiv g }$ on ${V \cap W}$. 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 ${\gamma = (f,V)}$, we can consider the totality ${X}$ of all equivalence classes of function elements that can be obtained by continuing ${\gamma}$ along curves in ${\mathbb{C}}$. Then ${X}$ is actually a Riemann surface. (more…)