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} <X, n> dS ,} (more…)

Just for fun, let’s see what all this actually means in local coordiantes.  It’s a good idea to get our hands dirty too.  To me at least, this makes the operators seem somewhat more friendly.

Choose local coordinates {x^1, \dots, x^n}. Let

\displaystyle g_{ij}(x) = < \partial_i, \partial_j >.

Let {g = \det( g_{ij})}. Note that {g>0}.

Henceforth, we will use the Einstein summation convention: all repeated indices are to be summed over, unless otherwise stated. For instance, the inner product is the 2-tensor

\displaystyle \frac{1}{2} g_{ij} dx^i \otimes dx^j.


Now let {X = X^j \partial_j} be a vector field. We want to compute {\mathrm{div} X} in terms of the quantities {X^i} and {g}. First, it will be necessary to compute the form {dV}. I claim that, if we take {x^i} to be oriented coordinates, then

\displaystyle dV = g^{1/2} dx^1 \wedge \dots \wedge dx^n. (more…)

I’ve set a tentative goal of heading towards the solution of the Dirichlet problem on compact manifoldw-with-boundary and Hodge theory; these will require various preliminaries, since of course it will be more fun to do it this way than to restrict to open sets in \mathbb{R}^n.  Today I will go through some of the basics of how the well-known operators from multivariable calculus work more generally on a Riemannian manifold, which will be necessary in the sequel.   (I shamelessly took the title of the post from a book I haven’t read.) 

Recall the well-known operator on functions of {n}-variables, the Laplacian

\displaystyle \Delta f := \sum_i \frac{ \partial^2f}{\partial x_i^{2}}.

The problem is, {\Delta} doesn’t transform nicely with respect to changes in coordinates, and since we want to define the Laplacian on manifolds, this causes a problem. However, it can be done on a Riemannian manifold. The idea is to use the formula

\displaystyle \Delta = \mathrm{div} (\mathrm{grad} ),

which is immediate from the definitions on {\mathbb{R}^n}. The key point is that {\mathrm{div} } and {\mathrm{grad}} make sense on any Riemannian manifold—from, respectively, vector fields to functions and from functions to vector fields. (more…)