Eigenfunctions of eigenvalue N2 are similar to polynomials of degree N, and one expects their zero sets to be similar to those of real algebraic varieties of degree N. I will discuss some classic and new results on the hypersurface volume of nodal sets which reflect this expectation and on the number of nodal domains (joint work in part with J. Toth and J. Jung as well as recent results of Ghosh-Reznikov-Sarnak).
By the size of eigenfunctions is meant their Lp norms. I will discuss some joint work with C. Sogge relating the size of eigenfunctions to the geometry of geodesic loops.
The relations between nodal sets and norms with classical mechanics are stronger if we complexify the manifold, analytically continue the eigenfunctions and study complex zeros. The complexification of M is the phase space of the mechanical system and the complex zeros reflect the ergodicity or integrability of the dynamics.