The Kato Square Root Problem on Submanifolds

TL;DR

This paper extends the solution of the Kato square root problem to divergence form operators on certain Riemannian submanifolds, using local quadratic estimates and Dirac operator frameworks.

Contribution

It provides a novel approach to the Kato problem on submanifolds with bounded second fundamental form by establishing local quadratic estimates for perturbed Dirac-type operators.

Findings

Solved the Kato square root problem on submanifolds with bounded second fundamental form.

Established local quadratic estimates for perturbations of Dirac-type operators.

Applicable to manifolds with exponential volume growth and local Poincaré inequality.

Abstract

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.