Local version of Courant's nodal domain theorem

TL;DR

This paper establishes a sharp local estimate for the number of nodal domains of Laplace eigenfunctions intersecting a given ball on a closed Riemannian manifold, advancing understanding of eigenfunction behavior.

Contribution

It introduces a novel local bound for nodal domains using a combination of Donnelly-Fefferman's idea, a Remez inequality, and a Landis growth lemma.

Findings

Sharp local bounds for nodal domains intersecting a ball

Extension of Donnelly-Fefferman's approach with new inequalities

Improved understanding of eigenfunction nodal structures

Abstract

Let $(M, g)$ be a closed Riemannian manifold, where g is $C^1$-smooth metric. Consider the sequence of eigenfunctions $u_k$ of the Laplace operator on M. Let $B$ be a ball on $M$. We prove a sharp estimate of the number of nodal domains of $u_k$ that intersect $B$. The problem of local bounds for the volume and for the number of nodal domains was raised by Donnelly and Fefferman, who also proposed an idea how one can prove such bounds. We combine their idea with two ingredients: the recent sharp Remez type inequality for eigenfunctions and the Landis type growth lemma in narrow domains.