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…)