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Rough metrics on manifolds and quadratic estimates

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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 “rough” 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 and obtain solutions to this problem on functions and forms on compact manifolds with a rough 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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Acknowledgments

This work was made possible through the support of the Australian-American Fulbright Commission through a Fulbright Scholarship, the Australian National University by the way of a supplementary Ph.D. scholarship, the Australian Government through an Australian Postgraduate Award, and the Australian Research Council through Discovery Project DP120103692 of Alan McIntosh and Pierre Portal. The author wishes to acknowledge the support of these organisations as well as both Alan and Pierre for their continuing support. Steve Zelditch deserves a special mention as this paper was motivated by a conversation with him. The author also wishes to acknowledge Rick Schoen, his Stanford Fulbright mentor, for his useful insights and feedback. Also, Kyler Siegel, Nick Haber, Boris Hanin, Mike Munn, Alex Amenta, Ziping Rao and Travis Willse deserve a mention for indulging the author in invigorating conversations. The first and less ambitious version of this paper was scrapped and re-written after a conversation with Sebastien Lucie. Annegret Burtscher deserves a special mention for her helpful insights and for offering corrections. Also, Anton Petrunin, Sergei V. Ivanov and Otis Chodosh need to be acknowledged for their insights, comments and contributions to the final section of this paper.

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  1. Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96, Gothenburg, Sweden

    Lashi Bandara

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  1. Lashi Bandara
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Correspondence to Lashi Bandara.

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Bandara, L. Rough metrics on manifolds and quadratic estimates. Math. Z. 283, 1245–1281 (2016). https://doi.org/10.1007/s00209-016-1641-x

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