I record a proof of the Faber-Krahn inequality here, mainly for my own benefit.

Let be one of the standard space forms: the Euclidean space , the unit sphere , or the hyperbolic space . Suppose is a bounded domain in with smooth boundary which is a closed hypersurface. Then we can define to be the geodesic ball in which has the same volume as .

Let be the first Laplacian eigenvalue of under the Dirichlet boundary condition. i.e. is the smallest number such that there exists a non-zero smooth function with

Theorem 1 (Faber-Krahn inequality)Suppose is a bounded domain in with smooth boundary . Then

The equality holds if and only if is a geodesic ball.

**Proof**: Let . Suppose is the first eigenfunction on . Symmetrize by the following procedure: define to be the radial function such that for any ,

It is easy to see that is constant on the level set of , so we have

By the isoperimetric inequality,

By the co-area formula,

As

by the construction , so in view of (2), we must have

By (1), (3) and minmax principle,

* By the isoperimetric inequality, the equality holds if and only if is a geodesic ball. *