SUBMANIFOLDS THAT ARE LEVEL SETS OF SOLUTIONS TO A SECOND-ORDER ELLIPTIC PDE

Abstract. We consider the problem of characterizing which noncompact hy-persurfaces in Rn can be regular level sets of a harmonic function modulo a C∞ diffeomorphism, as well as certain generalizations to other PDEs. We prove a versatile sufficient condition that shows, in particular, that any nonsingular al-gebraic 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 Rn. The technique we use, which is a significant re-finement of the basic strategy we recently applied to construct solutions to the Euler equation with knotted stream lines (Ann. of Math. 175 (2012) 345–367), 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. 1