Submanifolds that are level sets of solutions to a second-order elliptic PDE

We consider the problem of characterizing which noncompact hypersurfaces in RnRn can be regular level sets of a harmonic function modulo a C∞C∞ diffeomorphism, as well as certain generalizations to other PDEs. We prove a versatile sufficient condition that shows, in particular, that any nonsingular algebraic hypersurface whose connected components are all noncompact can be transformed onto a union of components of the zero set of a harmonic function via a diffeomorphism of RnRn. The technique we use combines robust but not explicit local constructions with appropriate global approximation theorems. In view of applications to a problem posed by Berry and Dennis, intersections of level sets are also studied.