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…)