Title: Growth and nodal sets of Laplace eigenfunctions on manifolds

Abstract:

We will discuss a new result 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 semi-classical limit. We also will discuss results that link the size of the nodal set to the growth of solutions of planar Schrodinger equations with small potential.