[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
[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
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 3 For 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 4 If is a complete non-compact simply-connected Riemannian manifold with sectional curvature , then
Remark 1 This 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 ,
If this is true, then the result would follow immediately by Cheeger’s theorem.
Lemma 5 is 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,
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
Consider , then we have and
Then we have
As at , we have
But then (3) implies . Our claim is proved.