Paris/Berkeley/Bonn/Zürich Analysis Seminar: Lower bounds on eigenfunctions on hyperbolic surfaces
Seminar | January 25 | 9:10-10 a.m. | 238 Sutardja Dai Hall
Semyon Dyatlov, UC Berkeley and MIT
I show that on a compact hyperbolic surface, the mass of an $L^2$-normalized eigenfunction of the Laplacian on any nonempty open set is bounded below by a positive constant depending on the set, but not on the eigenvalue. This statement, more precisely its stronger semiclassical version, has many applications including control for the Schrödinger equation and the full support property for semiclassical defect measures. The key new ingredient of the proof is a fractal uncertainty principle, stating that no function can be localized close to a porous set in both position and frequency. This talk is based on joint works with Long Jin and with Jean Bourgain.