Number of nodal domains and singular points of eigenfunctions of negatively\n\t\t\t\tcurved surfaces with an isometric involution

Abstract

We prove two types of nodal results for density $1$ subsequences of an\northonormal basis $\\{ \\phi_j \\}$ of eigenfunctions of the Laplacian on a\nnegatively curved compact surface $(M,g)$. The first result pertains to Riemann\nsurfaces $(M, J, \\sigma)$ with an anti-holomorphic involution $\\sigma$ such that\n$M - \\mathrm{Fix}(\\sigma)$ has more than one component. In any genus\n$\\mathfrak{g}$, there is a $(3\\mathfrak{g} - 3)$-dimensional moduli space of\nsuch real Riemann surfaces. Our main result is that, for any negatively curved\n$\\sigma$-invariant metric $\\mathfrak{g}$ on $M$, the number of nodal domains of\nthe even or odd $\\Delta_{\\mathfrak{g}}$-eigenfunctions tends to infinity along a\ndensity $1$ subsequence. For a generic $\\sigma$-invariant negatively curved\nmetric $\\mathfrak{g}$, the multiplicity of all eigenvalues equals $1$, and all\neigenfunctions are either even or odd, and therefore the result holds for almost\nany eigenfunction.\n¶ The analytical part of the proof shows that the number of zeros of even\neigenfunctions restricted to $\\mathrm{Fix}(\\sigma)$, and the number of singular\npoints of odd eigenfunctions on $\\mathrm{Fix}(\\sigma)$, tend to infinity. This\nis a quantum ergodic restriction phenomenon. Our second result generalizes this\nstatement to any negatively curved surface $(M,g)$ and to a generic curve $C\n\\subset M$: the number of zeros of eigenfunctions $\\phi_j |_C$ tends to\ninfinity.\n¶ The additional step to obtain a growing number of nodal domains in the $(M, J,\n\\sigma)$ setting is topological. It generalizes an argument of Ghosh, Reznikov,\nand Sarnak on the modular domain to higher genus.