Boundaries of Graphs of Harmonic Functions

TL;DR

This paper characterizes the boundaries of harmonic functions' graphs using conservation laws, providing necessary and sufficient conditions for boundary existence and offering a new proof for holomorphic disk boundaries.

Contribution

It introduces a novel boundary characterization for harmonic functions via conservation laws and proves sufficiency using elliptic theory, with applications to holomorphic disks.

Findings

Necessary conditions for boundary existence are also sufficient locally.

Standard elliptic theory can construct integral manifolds with prescribed boundaries.

Provides a new elementary proof for boundaries of holomorphic disks.

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.