Some remarks on spherical harmonics

TL;DR

This paper explores properties of spherical harmonics, including zero set constructions, bounds on nodal lengths, and intersections, providing new insights into their geometric and combinatorial structure.

Contribution

It introduces new constructions, bounds, and formulas for the zero sets and intersections of spherical harmonics, advancing understanding of their geometric properties.

Findings

Sharp upper bounds for nodal length

Precise bounds for common zeroes of two harmonics

Mean Hausdorff measure of intersections of nodal sets

Abstract

The article contains several observations on spherical harmonics and their nodal sets: a construction for harmonics with prescribed zeroes; a kind of canonical representation of this type for harmonics on $\bbS^2$; upper and lower bounds for nodal length and inner radius (the upper bounds are sharp); precise upper bound for the number of common zeroes of two spherical harmonics on $\bbS^2$; the mean Hausdorff measure on the intersection of $k$ nodal sets for harmonics of different degrees on $\bbS^m$, where $k\leq m$ (in particular, the mean number of common zeroes of $m$ harmonics).