Log-scale equidistribution of nodal sets in Grauert tubes

TL;DR

This paper proves that on Grauert tubes of negatively curved manifolds, the complexified eigenfunctions and their zeros become equidistributed at a logarithmic scale along a full density subsequence of eigenvalues.

Contribution

It establishes the first logarithmic scale equidistribution results for complexified eigenfunctions and their zeros on Grauert tubes of negatively curved manifolds.

Findings

Complexified eigenfunctions are asymptotically equidistributed at logarithmic scales.

Zeros of complexified eigenfunctions become equidistributed at the same logarithmic scales.

Results hold for a full density subsequence of eigenvalues.

Abstract

Let $M_{\tau_0}$ be the Grauert tube (of some fixed radius $\tau_0$) of a compact, negatively curved, real analytic Riemannian manifold $M$ without boundary. Let $\phi_\lambda$ be a Laplacian eigenfunction on $M$ of eigenvalues $-\lambda^2$ and let $\phi_\lambda^\mathbb{C}$ be its holomorphic extension to $M_{\tau_0}$. In this article, we prove that on $M_{\tau_0} \setminus M$, there exists a dimensional constant $\alpha > 0$ and a full density subsequence $ \{\lambda_{j_k}\}_{k=1}^{\infty}$ of the spectrum for which the masses of the complexified eigenfunctions $\phi_{\lambda_{j_k}}^\mathbb{C}$ are asymptotically equidistributed at length scale $(\log \lambda_{j_k})^{-\alpha}$. Moreover, the complex zeros of $\phi_{\lambda_{j_k}}^\mathbb{C}$ also become equidistributed on this logarithmic length scale.

Equations

$$(\Delta +{\lambda }_j^2){\phi }_j=0,$$

$$M_{\tau }:=\{\zeta \in M_{\mathbb{C}}:\sqrt{\rho }(\zeta )<\tau \}$$

$$d\mu :={\omega }^n\text{and}d{\mu }_{\tau }:=\frac{{\omega }^n}{d\sqrt{\rho }{\mid }_{\partial M_{\tau }}}=\frac{{\omega }^n}{d\tau }.$$

$${\mathcal{Z}}_j:=\{\zeta \in M_{{\tau }_0}:{\phi }_j^{\mathbb{C}}(\zeta )=0\}.$$

$$⟨[{\mathcal{Z}}_j],\eta ⟩:={\int }_{{\mathcal{Z}}_j}\eta ={\int }_{M_{{\tau }_0}}\frac{i}{2\pi }\partial \bar{\partial }\log |{\phi }_j^{\mathbb{C}}{|}^2\wedge \eta$$

$$⟨|{\mathcal{Z}}_j|,f⟩:={\int }_{{\mathcal{Z}}_j}f{\omega }^{n-1},f\in C(M_{{\tau }_0}).$$

$$\frac{1}{{\lambda }_{j_k}}[{\mathcal{Z}}_{j_k}]\rightharpoonup \frac{i}{\pi }\partial \bar{\partial }\sqrt{\rho }\text{weakly as currents on Mτ0}$$

λjk​​1​[Zjk​​]⇀πi​∂∂ˉ