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 .
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.
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,
by the construction , so in view of (2), we must have
By the isoperimetric inequality, the equality holds if and only if is a geodesic ball.