Nested nodal loops for sums of Laplace eigenfunctions

TL;DR

This paper investigates the structure of zero sets of Laplace eigenfunction sums on surfaces, establishing bounds on nested loops in real-analytic cases and providing counterexamples in smooth cases, with implications for biharmonic functions.

Contribution

It proves a uniform bound on nested loops in real-analytic eigenfunctions, constructs a biharmonic function with a double nest, and explores the limits of these phenomena.

Findings

Bound on rooted double nests for real-analytic eigenfunctions

Existence of infinitely many nests on smooth spheres with specific eigenfunction combinations

Construction of a biharmonic function with a double nest in its nodal set

Abstract

We study nested loops in zero sets of sums of Laplace eigenfunctions on closed surfaces. In the real-analytic category, answering a question of Logunov, we prove a uniform bound for the number of rooted double nests in terms of the surface, the root, and the spectral cutoff. We show that this analyticity hypothesis is sharp: on a smooth sphere, a linear combination of eigenfunctions with eigenvalues \(0\) and \(2\) can have infinitely many rooted double nests. We also answer a question of Logunov and Nadirashvili by constructing a planar biharmonic function whose nodal set contains a double nest, and we prove a quantitative bound for entire biharmonic functions of polynomial growth. The biharmonic construction gives a nodal-set manifestation of the failure of the Boggio--Hadamard conjecture from the 1900s.