The Kato Square Root Problem on Submanifolds

We solve the Kato square root problem for divergence form operators on complete Riemannian manifolds that are embedded in Euclidean space with a bounded second fundamental form. We do this by proving local quadratic estimates for perturbations of certain first-order differential operators that act on the trivial bundle over a complete Riemannian manifold with at most exponential volume growth and on which a local Poincar\'{e} inequality holds. This is based on the framework for Dirac type operators that was introduced by Axelsson, Keith and McIntosh.