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 σ 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.

Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza–Klein 3-folds

“…To put the nodal results into context, it is proved in varying degrees of generality in [8,9,11,12,13,14,17,22] that in dimension 2, the number of nodal domains of an orthonormal basis {u j } of Laplace eigenfunctions on certain surfaces with ergodic geodesic flow tends to infinity with the eigenvalue along almost the entire sequence of eigenvalues. By the first item of Theorem 1.5, the same is true for their lifts to the unit tangent bundle SX as invariant eigenfunctions of the Kaluza-Klein metric.…”

Small Scale Equidistribution of Random Eigenbases

“…Note that the range of α in (ii) was improved to α ∈ [0, 1/(2n)) (same as in (i)) by Hezari-Rivière [HR1]. The equidistribution of eigenfunctions at logarithmic scales has since been applied to the L p norm and nodal set estimates of eigenfunctions by Hezari-Rivière [HR1] and Sogge [So2] and counting nodal domains of eigenfunctions by Zelditch [Ze3]. See also a recent survey by Sogge [So3].…”

Nodal set of monochromatic waves satisfying the Random Wave Model

“…Jang and Jung [32] obtained unconditional results for individual Hecke-Maass eigenfunctions of arithmetic triangle groups. Jung and Zelditch [33] proved, generalising the geometric argument in [25,24], that N (•) tends to infinity, for most eigenfunctions on certain negatively curved manifolds, and Zelditch [53] gave a logarithmic lower bound. Finally, Ingremeau [29] gave examples of eigenfunctions with N (•) → ∞ on unbounded negatively-curved manifolds.…”