Paper detail

Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution

We prove two types of nodal results for density one subsequences of an orthonormal basis $\{ϕ_j\}$ of eigenfunctions of the Laplacian on a negatively curved compact surface. The first type of result involves the intersections $Z_{ϕ_j} \cap H$ of the nodal set $Z_{ϕ_j}$ of $ϕ_j$ with a smooth curve $H$. Using recent results on quantum ergodic restriction theorems and prior results on periods of eigenfunctions over curves, we prove that under an asymmetry assumption on $H$, the number of intersection points tends to infinity for a density one subsequence of the $ϕ_j$. . We also prove that the number of zeros of the normal derivative $\partial_ν ϕ_j$ on $H$ tends to infinity. From these results we obtain a lower bound on the number of nodal domains of even and odd eigenfunctions on surfaces with an isometric involution. Using (and generalizing) a geometric argument of Ghosh-Reznikov-Sarnak, we show that the number of nodal domains of even or odd eigenfunctions tends to infinity for a density one subsequence of eigenfunctions.