We prove a result, announced by F. Nazarov, L. Polterovich and M. Sodin, that
exhibits a relation between the average local growth of a Laplace eigenfunction on a
closed surface and the global size of its nodal set. More precisely, we provide a lower
and an upper bound to the Hausdorff measure of the nodal set in terms of the
expected value of the growth exponent of an eigenfunction on disks of wavelength-like
radius. Combined with Yau’s conjecture, the result implies that the average local
growth of an eigenfunction on such disks is bounded by constants in the
semiclassical limit. We also obtain results that link the size of the nodal
set to the growth of solutions of planar Schrödinger equations with small
potential.
Keywords
spectral geometry, Laplace eigenfunctions, nodal sets,
growth of eigenfunctions
