Three Topological Properties of Small Eigenfunctions on Hyperbolic Surfaces
Jean-Pierre Otal · Birkhäuser Basel eBooks · 2008
We apply topological methods for studying eigenfunctions on finite volume hyperbolic surfaces. From the Lemma saying that any non-zero eigenfunction on an hyperbolic surface with eigenvalue ≤ 1/4 has an incompressible nodal set, we deduce the following propositions: 1) the non-existence of cuspidal eigenfunctions with eigenvalue ≤ 1/4 on surfaces of genus 0 or 1 (a result already obtained by Huxley); 2) the dimension of a cuspidal eigenspace with eigenvalue ≤ 1/4 is not more than 2 g −3 (a generalization of 1)); 3) a dichotomy for functions in an eigenspace with eigenvalue ≤ 1/4: either an eigenfunction exists which has a smooth nodal set / or all functions in the eigenspace have common zeroes.