Boundaries of Graphs of Harmonic Functions

Harmonic functions $u\colon\mathbb R^n\to\mathbb R^m$ are equivalent to integral manifolds of an exterior differential system with independence condition $(M,\mathcal T,\omega)$. To this system one associates the space of conservation laws $\mathcal C$. They provide necessary conditions for $g\colon\mathbb S^{n-1}\to M$ 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 $g(\mathbb S^{n-1})$. The proof uses standard linear elliptic theory to produce an integral manifold $G\colon D^n\to M$ and the completeness of the space of conservation laws to show that this candidate has $g(\mathbb S^{n-1})$ as its boundary. As a corollary we obtain a new elementary proof of the characterization of boundaries of holomorphic disks in $\mathbb C^m$ in the local case.