Metrics with large first eigenvalue

I will start by recalling some basic facts on the first eigenvalue of
the Laplacian on a compact Riemannian manifold. Next I will discuss a
classical theorem of Hersch, which says that on the 2-sphere the first
eigenvalue is the largest possible when the metric is the standard
one.

I will then formulate the analogous problem for Kaehler metrics and
will describe the solution in the case of Hermitian symmetric
spaces of the compact type. If time permits I will try to explain the
link with Satake-Furstenberg compactifications and coadjoint orbits.