One of the basic properties of the Laplacian is that given a compact Riemannian manifold-with-boundary (to which all this ${\mathrm{div}, \mathrm{grad}, \Delta}$ business applies equally), then for ${u}$ vanishing on the boundary, the ${L^2}$ inner product ${(\Delta u, u)}$ is fairly large relative to ${u}$. As an immediate corollary, if ${u}$ satisfies the Laplace equation ${\Delta u= 0}$ and vanishes on the boundary, then ${u}$ is identically zero.

It turns out that the proof of this will require the divergence theorem. This is a familiar fact from multivariable calculus, but it generalizes to ${n}$-dimensions nicely as a corollary of Stokes theorem and some of the other machinery thus developed.

So, let’s choose an oriented Riemannian manifold ${M}$ of dimension ${n}$ with boundary ${\partial M}$. There is a volume form ${dV}$ because of the choice of orientation globally defined. On ${\partial M}$, there is an induced Riemannian metric and an induced orientation, with a corresponding volume form ${dS}$ on ${\partial M}$. If ${X}$ is a compactly supported vector field, the divergence theorem states that

$\displaystyle \boxed{ \int_M \mathrm{div} X dV = \int_{\partial M} dS ,}$ (more…)

A friend of mine is taking a course on analytic number theory in the spring and needs to learn basic complex analysis in a couple of weeks.  I decided to do a post (self-contained, except for Stokes’ formula) on deducing the Cauchy theorems and their applications from Stokes’ theorem now instead of later–when I’ll talk about several complex variables.  It might be objected that Stokes’ theorem is just Green’s theorem for $n=2$, commonly used in undergraduate treatments, but my goal was to take an expository challenge: write something rigorous on complex variables in as short a space as possible without sacrificing readability.  So Stokes’ theorem for manifolds is preferable to Green’s theorem as stated in a vague way about “insides of a curve” (before, say, the Jordan curve theorem is proved) and the traditional proof of Green’s theorem via rectangular decompositions.

So, let’s consider an open set ${O \subset \mathbb{C}}$, and a ${C^2}$ function ${f: O \rightarrow \mathbb{C}}$. We can consider the differential

$\displaystyle df := f_x dx + f_y dy$

which is a complex-valued 1-form on ${O}$. It is also convenient to write the differential using the ${z}$ and ${\bar{z}}$-derivatives I talked about earlier, i.e.

$\displaystyle f_z := \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) f, \quad f_{\bar{z}} := \frac{1}{2}\left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) f.$

The reason these are important is that if ${w_0 \in O}$, we can choose ${A,B \in \mathbb{C}}$ with

$\displaystyle f(w_0+h) = f(w_0) + Ah + B \bar{h} + o(|h|), \ h \in \mathbb{C}$

by differentiability, and it is easy to check that ${A=f_z(w_0), B=f_{\bar{z}}(w_0)}$. So we can define a function ${f}$ to be holomorphic if it satisfies the differential equation

$\displaystyle f_{\bar{z}} = 0,$

which is equivalent to being able to write

$\displaystyle f(w_0 + h) =f(w_0) + Ah + o(|h|)$

for each ${w_0 \in O}$ and a suitable ${A \in \mathbb{C}}$. In particular, it is equivalent to a difference quotient definition. The derivative ${f_z}$ of a holomorphic function thus satisfies all the usual algebraic rules, under which holomorphic functions are closed. (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…)