Log lower bound on the number of nodal domains on some surfaces of negative curvature

Abstract: An open problem is to prove that for any (or at least any generic) Riemannian metric, there is some sequence of eigenfunctions of the Laplacian for which the number of nodal domains tends to infinity. It sounds easy but as yet there are almost no examples of such metrics except for separation of variables situations, and the results of Ghosh-Reznikov-Sarnak and of Junehyuk Jung and myself. This talk is about a quantitative improvement in which we give a log lower bound for the number of nodal domains for negatively curved `real Riemann surfaces'. The same result should hold at least for any non-positively curved surface with concave boundary, but there are several technical obstructions.