We prove that the Atiyah-Singer Dirac operator / D g in L 2 depends Riesz continuously on L ∞ perturbations of complete metrics g on a smooth manifold. The Lipschitz bound for the map g → / D g (1 + / D
Abstract. We consider smooth, complete Riemannian manifolds which are exponentially locally doubling. Under a uniform Ricci curvature bound and a uniform lower bound on injectivity radius, we prove a Kato square root estimate for certain coercive operators over the bundle of finite rank tensors. These results are obtained as a special case of similar estimates on smooth vector bundles satisfying a criterion which we call generalised bounded geometry. We prove this by establishing quadratic estimates for perturbations of Dirac type operators on such bundles under an appropriate set of assumptions.
Abstract. We study some canonical differential operators on vector bundles over smooth, complete Riemannian manifolds. Under very general assumptions, we show that smooth, compactly supported sections are dense in the domains of these operators. Furthermore, we show that smooth, compactly supported functions are dense in second order Sobolev spaces on such manifolds under the sole additional assumption that the Ricci curvature is uniformly bounded from below.
Abstract. We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of Riemannian-like metrics that are permitted to be of low regularity and degenerate on sets of measure zero. We also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi-Lipschitz. As a consequence, we demonstrate the invariance of the Kato square root problem under Lipschitz transformations of the space and obtain solutions to this problem on functions and forms on compact manifolds with a continuous metric. Furthermore, we show that a lower bound on the injectivity radius is not a necessary condition to solve the Kato square root problem.
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