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

We show for a certain class of operators A and holomorphic functions f that the functional calculus A bar right arrow 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.

Automated summary

We prove the holomorphy of the functional calculus A bar right arrow f (A) for a certain class of operators A and holomorphic functions f. Using this result, we demonstrate the real analytic dependence of fractional Laplacians (1 + Delta(g))(p) on the metric g in suitable Sobolev topologies. As an application, we establish the local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics.