  We estimate the volume of superlevel sets of Laplace-Beltrami eigenfunctions
on a compact Riemannian manifold. The proof uses the Green's function
representation and the Bathtub principle. As an application, we obtain upper
bounds on the distribution of the extrema of a Laplace-Beltrami eigenfunction
over its nodal domains. Such bounds have been previously proved by L.
Polterovich and M. Sodin in the case of compact surfaces. Our techniques allow
to generalize these results to arbitrary dimensions. We also discuss a
different approach to the problem based on reverse H\"older inequalities due to
G. Chiti.

Spectral Theory