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Smooth Perturbations of the Functional Calculus and Applications to Riemannian Geometry on Spaces of Metrics

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Abstract

We show for a certain class of operators A and holomorphic functions f that the functional calculus \(A\mapsto f(A)\) is holomorphic. Using this result we are able to prove that fractional Laplacians \((1+\Delta ^g)^p\) depend real analytically on the metric g in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics.

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Authors and Affiliations

  1. Department of Mathematics, Florida State University, Tallahassee, USA

    Martin Bauer

  2. Onfido, 3 Finsbury Avenue, London, EC2M 2PA, UK

    Martins Bruveris

  3. Division of Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Philipp Harms

  4. Faculty for Mathematics, University of Vienna, Wien, Austria

    Peter W. Michor

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  1. Martin Bauer
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  2. Martins Bruveris
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Communicated by P. Chrusciel.

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We thank Lashi Bandara, Andreas Kriegl, Peer Kunstmann, Gerard Misiolek, Marvin S. Müller, Armin Rainer, and Lutz Weis for helpful discussions. MB was partially supported by NSF-grants 1912037 and 1953244. PH was partially supported in the form of a Junior Fellowship of the Freiburg Institute of Advances Studies.

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Bauer, M., Bruveris, M., Harms, P. et al. Smooth Perturbations of the Functional Calculus and Applications to Riemannian Geometry on Spaces of Metrics. Commun. Math. Phys. 389, 899–931 (2022). https://doi.org/10.1007/s00220-021-04264-y

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