Logarithmic lower bound on the number of nodal domains

Abstract

We prove that the number of nodal domains of eigenfunctions grows at least logarithmically with the eigenvalue (for almost the entire sequence of eigenvalues) on certain negatively curved surfaces. The geometric model is the same as in prior joint work with J. Jung, where the number of nodal domains was shown to tend to infinity. The surfaces are assumed to be “real Riemann surfaces,” i.e. Riemann surfaces with an anti-holomorphic involution with non-empty fixed point set. The eigenfunctions are assumed to be even or odd, which is automatically the case for generic invariant metrics. The logarithmic growth rate gives a quantitative refinement of the prior results.