Lashi Bandara

The Kato Square Root Problem on Vector Bundles with Generalised Bounded Geometry

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.

Square roots of perturbed subelliptic operators on Lie groups

The authors appreciate the support of the Centre for Mathematics and its Applications at the Australian National University, Canberra, where this project was begun, the Department of Mathematics at the University of Auckland, and the Department of Mathematics, University of Missouri, where it was continued. The first author was supported through an Australian Postgraduate Award, through the Mathematical Sciences Institute, Australian National University, Canberra, and an Australian-American Fulbright Scholarship. The second and third authors were supported by the Marsden Fund Council from N.Z. Government funding, administered by the Royal Society of New Zealand. The third author gratefully acknowledges support from the Australian Government through the Australian Research Council.

Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of the metric

We prove that the Atiyah–Singer Dirac operator in

Essential self-adjointness of powers of first-order differential operators on non-compact manifolds with low-regularity metrics

\mathrm{L}^{2}_\mathrm{}

Self-adjointness of the Gaffney Laplacian on Vector Bundles

L2 depends Riesz continuously on

Geometric singularities and a flow tangent to the Ricci flow

\mathrm{L}^{\infty }_\mathrm{}