  A differential operator $T$ satisfies the $L^2$-unique continuation property
if every $L^2$-solution of $T$ that vanishes on an open subset vanishes
identically. We study the $L^2$-unique continuation property of an operator $T$
acting on a manifold with bounded geometry. In particular, we establish some
connections between this property and the regularity properties of $T$. As an
application, we prove that the deformation operator on a manifold with bounded
geometry satisfies regularity and $L^2$-unique continuation properties. As
another application, we prove that suitable elliptic operators are invertible
(Hadamard well-posedness). Our results apply to compact manifolds, which have
bounded geometry.
