Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics Article Swipe

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Geodesic Sobolev space Mathematics Holomorphic function Riemannian geometry Fractional calculus Pure mathematics Metric (unit) Space (punctuation) Functional calculus Solving the geodesic equations Class (philosophy) Mathematical analysis Topology (electrical circuits) Computer science Combinatorics Artificial intelligence Operating system Economics Operations management

Martin Bauer , Martins Bruveris , Philipp Harms , Peter W. Michor ·

YOU? · · 2019 · Open Access · · DOI: https://doi.org/10.14288/1.0379393 · OA: W2895095849

We show that the functional calculus, which maps operators A to functionals f(A), is holomorphic for a certain class of operators A and holomorphic functions f. Using this result we are able to prove that fractional Laplacians depend real analytically on the underlying Riemannian metric 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. (Joint work with Martins Bruveris, Martin Bauer, and Peter W. Michor)