Quadratic Cones on Which Few Harmonic Functions Vanish

This paper is concerned with the space of harmonic functions vanishing on a given subset of Euclidean space. In dimensions three and higher, we show that for the cone cut out by a generically chosen harmonic quadratic polynomial this space is exactly two-dimensional. This phenomenon allows the following generalization to arbitrary elliptic differential operators of second order: Consider the level set of a solution at a nondegenerate critical value. As long as the tangent cone to the level set at a critical point satisfies a certain genericity condition, the space of solutions vanishing on the level set is at most two-dimensional.