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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 resultwe are able to prove that fractional Laplacians $(1+\Delta^g)^p$ depend realanalytically on the metric $g$ in suitable Sobolev topologies. As anapplication we obtain local well-posedness of the geodesic equation forfractional Sobolev metrics on the space of all Riemannian metrics.

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