Boundaries of Graphs of Harmonic Functions

Harmonic functions u : R → R are equivalent to integral manifolds of an exterior differential system with independence condition (M, I, ω). To this system one associates the space of conservation laws C. They provide necessary conditions for g : Sn−1 →M to be the boundary of an integral submanifold. We show that in a local sense these conditions are also sufficient to guarantee the existence of an integral manifold with boundary g(Sn−1). The proof uses standard linear elliptic theory to produce an integral manifold G : D → M and the completeness of the space of conservation laws to show that this candidate has g(Sn−1) as its boundary. As a corollary we obtain a new elementary proof of the characterization of boundaries of holomorphic disks in C in the local case.