TY - JOUR

T1 - Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution

AU - Jung, Junehyuk

AU - Zelditch, Steve

PY - 2016/1

Y1 - 2016/1

N2 - We prove two types of nodal results for density 1 subsequences of an orthonormal basis {φj} of eigenfunctions of the Laplacian on a negatively curved compact surface (M,g). The first result pertains to Riemann surfaces (M,J, σ) with an anti-holomorphic involution σ such that M -Fix(σ) has more than one component. In any genus g, there is a (3g-3)-dimensional moduli space of such real Riemann surfaces. Our main result is that, for any negatively curved σ-invariant metric g on M, the number of nodal domains of the even or odd Δg-eigenfunctions tends to infinity along a density 1 subsequence. For a generic σ-invariant negatively curved metric g, the multiplicity of all eigenvalues equals 1, and all eigenfunctions are either even or odd, and therefore the result holds for almost any eigenfunction. The analytical part of the proof shows that the number of zeros of even eigenfunctions restricted to Fix(σ), and the number of singular points of odd eigenfunctions on Fix(σ), tend to infinity. This is a quantum ergodic restriction phenomenon. Our second result generalizes this statement to any negatively curved surface (M,g) and to a generic curve C ⊂ M: The number of zeros of eigenfunctions φj |C tends to infinity. The additional step to obtain a growing number of nodal domains in the (M,J, σ) setting is topological. It generalizes an argument of Ghosh, Reznikov, and Sarnak on the modular domain to higher genus.

AB - We prove two types of nodal results for density 1 subsequences of an orthonormal basis {φj} of eigenfunctions of the Laplacian on a negatively curved compact surface (M,g). The first result pertains to Riemann surfaces (M,J, σ) with an anti-holomorphic involution σ such that M -Fix(σ) has more than one component. In any genus g, there is a (3g-3)-dimensional moduli space of such real Riemann surfaces. Our main result is that, for any negatively curved σ-invariant metric g on M, the number of nodal domains of the even or odd Δg-eigenfunctions tends to infinity along a density 1 subsequence. For a generic σ-invariant negatively curved metric g, the multiplicity of all eigenvalues equals 1, and all eigenfunctions are either even or odd, and therefore the result holds for almost any eigenfunction. The analytical part of the proof shows that the number of zeros of even eigenfunctions restricted to Fix(σ), and the number of singular points of odd eigenfunctions on Fix(σ), tend to infinity. This is a quantum ergodic restriction phenomenon. Our second result generalizes this statement to any negatively curved surface (M,g) and to a generic curve C ⊂ M: The number of zeros of eigenfunctions φj |C tends to infinity. The additional step to obtain a growing number of nodal domains in the (M,J, σ) setting is topological. It generalizes an argument of Ghosh, Reznikov, and Sarnak on the modular domain to higher genus.

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U2 - 10.4310/jdg/1452002877

DO - 10.4310/jdg/1452002877

M3 - Article

AN - SCOPUS:84955448745

VL - 102

SP - 37

EP - 66

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 1

ER -