Ruled common nodal surfaces
Nodal sets are zero loci of Laplace eigenfunctions (e.f.). Study of nodal sets is important
for understanding wave processes. The geometry of a single nodal set may be very complicated
and hardly can be well understood. More realistic might be describing geometry of sets which
are nodal for a large family of e.f. (the condition of simultaneous vanishing, resonanse, of
a large packet of e.f., on a large set, is overdetermined and hence may be expected to occur
only for exclusive sets).
Indeed, it was proved that common nodal curves for large, in different senses, families
of e.f. in $\mathbb R^2$ are straight lines (non-periodic case: Quinto and the speaker, ’96; periodic
case: Bourgain and Rudnick, ’11). It was conjectured that in a Euclidean space of
arbitrary dimension, common nodal hypersurfaces for large families of e.f. are cones, more precisely,
are translates of zero sets of harmonic homogeneous polynomials.
The talk will be devoted to a recent result confirming the conjecture for ruled hypersurfaces
in $\mathbb R^3$. Relation to the injectivity problem for the spherical Radon transform will be explained.