A unique continuation theorem for exterior differential forms on Riemannian manifolds with boundary

TL;DR

This paper extends a unique continuation theorem for differential forms to Riemannian manifolds with boundary, providing new boundary condition considerations and applications to zero set dimensions of harmonic and eigenfield forms.

Contribution

It generalizes a known unique continuation result to manifolds with boundary, incorporating boundary conditions and analyzing zero set dimensions of harmonic and eigenfield forms.

Findings

Zero set Hausdorff dimension bounds with codimension at least 2

Extension of unique continuation to manifolds with boundary

Applications to harmonic Neumann and Dirichlet forms, and curl eigenfields

Abstract

Aronszajn, Krzywicki and Szarski proved in \cite{AKS62} a strong unique continuation result for differential forms, satisfying a certain first order differential inequality, on Riemannian manifolds with empty boundary. The present paper extends this result to the setting of Riemannian manifold with non-empty boundary, assuming suitable boundary conditions on the differential forms. We then present some applications of this extended result. Namely, we show that the Hausdorff dimension of the zero set of harmonic Neumann and Dirichlet forms, as well as eigenfields of the curl operator (on $3$-manifolds), has codimension at least $2$. Again, these bounds were known in the setting of manifolds without boundary, so that the merit is once more the inclusion of boundary points.