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

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.

32 pages, minor revision

References in corpus (3)

• An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
• Vanishing distance phenomena and the geometric approach to SQG
• Fractional Sobolev metrics on spaces of immersions

Cited by in corpus (4)

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• Long-time existence of Brownian motion on configurations of two landmarks
• Regularity and completeness of half-Lie groups
• On Derkachov--Manashov $R$-matrices for the principal series of unitary representations