Quadratic cones on which few harmonic functions vanish

TL;DR

This paper demonstrates that in dimensions three and higher, the space of harmonic functions vanishing on certain quadratic cones is limited to two dimensions, a result that extends to general elliptic operators under generic conditions.

Contribution

It establishes a robust, general result about the dimensionality of harmonic functions vanishing on quadratic cones and level sets for elliptic operators, under generic conditions.

Findings

Harmonic functions vanishing on quadratic cones are at most two-dimensional in higher dimensions.

The result extends to elliptic operators with level sets satisfying genericity conditions.

The phenomenon is robust and generalizes beyond harmonic functions to broader elliptic PDEs.

Abstract

We show that, in dimension three and higher, the space of harmonic functions vanishing on the cone defined by a generically chosen harmonic quadratic polynomial is two-dimensional. This phenomenon is surprisingly robust, generalizing to arbitrary elliptic differential operators of second order, with the cone replaced by the level set of a solution at a nondegenerate critical value. As long as the tangent cone to the level set at the critical point satisfies a certain genericity condition, the space of solutions vanishing on the level set is at most two-dimensional.