[Update: a simplified proof (9-10-2011), Rauch comparison proved (12-10-2011) ]

For a compact Riemannian manifold with boundary , if the metric is , we define the Laplace operator

on where and . We define by the completion of with the norm

and can be extended as a self-adjoint non-positive definite operator from to . Let be the (Dirichlet) eigenvalues of , i.e. the eigenfunctions satisfy

[Physically, the eigenvalues can be thought of as the fundamental modes (all possible frequencies) if you imagine as a drum. ] We are interested to find out the first eigenvalue , or at least estimate it (lower bound is often more important) in terms of the geometric conditions. First we give a definition.

Definition 1 (Cheeger)For a compact Riemannian manifold with boundary , define the isoperimetric constant

An important result is

Theorem 2 (Cheeger)For a compact Riemannian manifold with boundary,

*Proof:* For an eigenfunction with respect to , integrating and applying divergence theorem, as ,

Now we claim that for any smooth function with ,

If this is true, then taking , we have

thus

which is what we want. It remains to prove our claim. We have to use the co-area formula: . Using this we have

where we have used the layer-cake representation formula in the last equality and the definition of is the last inequality.

Let us apply the above theorem to estimate the first eigenvalue for a Riemannian manifold which have a negative upper bound for its sectional curvature.

Definition 3For a complete noncompact simply-connected Riemannian manifold we define

where . Since the RHS of the above is non-increasing and is positive, the limit exists.

We now state a theorem of McKean.

Theorem 4If is a complete non-compact simply-connected Riemannian manifold with sectional curvature , then

Remark 1This estimate is sharp, in the sense that for the hyperbolic space (of curvature ) is exactly .

Since I don’t quite understand McKean’s original proof myself, I will give a modified proof here.

*Proof:* As is simply connected with negative curvature, by Cartan-Hadamard theorem, the distance function is differentiable on . So for ,

where is the outer normal of . We will show that

If this is true, then the result would follow immediately by Cheeger’s theorem.

Lemma 5is the second fundamental form of the geodesic sphere (at the point ) centered at with radius . In particular, is the mean curvature of .

*Proof:* For tangent vector fields to , as is the unit outward normal,

We now show that the eigenvalues of are , then by Lemma 5, (2) will follow immediately.To see this, we need the following

Proposition 6 (Rauch’s comparison theorem)With the same assumptions as in Theorem 4, for any tangent vector to at ,

It then follows that all the eigenvalues of are bounded from below by .

*Proof:* We can assume the tangent vector at . Let be the Jacobi field on with and . As , we have

So

Thus

Consider , then we have and

Then we have

As at , we have

But then (3) implies . Our claim is proved.