Quadratic cones on which few harmonic functions vanish

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.