Boundaries of Graphs of Harmonic Functions

Abstract

Harmonic functions $u:{\mathbb R}^n \to {\mathbb R}^m$ are equivalent to integral manifolds of an exterior differential system with independence condition $(M,{\mathcal I},\omega)$. To this system one associates the space of conservation laws ${\mathcal C}$. They provide necessary conditions for $g:{\mathbb S}^{n-1} \to M$ 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 $G:D^n \to M$ 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.