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

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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.

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Acknowledgments

The authors appreciate the support of the Centre for Mathematics and its Applications at the Australian National University, Canberra, where this project was undertaken. The first author was supported through an Australian Postgraduate Award and through the Mathematical Sciences Institute, Australian National University, Canberra. The second author gratefully acknowledges support from the Australian Government through the Australian Research Council. The authors thank Andrew Morris for helpful conversations and suggestions.

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  1. Centre for Mathematics and its Applications, Australian National University, Canberra, ACT, 0200, Australia

    Lashi Bandara & Alan McIntosh

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  1. Lashi Bandara
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  2. Alan McIntosh
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Correspondence to Lashi Bandara.

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Bandara, L., McIntosh, A. The Kato Square Root Problem on Vector Bundles with Generalised Bounded Geometry. J Geom Anal 26, 428–462 (2016). https://doi.org/10.1007/s12220-015-9557-y

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  • DOI: https://doi.org/10.1007/s12220-015-9557-y

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