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

TL;DR

This paper proves the holomorphic dependence of certain operator functions on metrics and applies this to show real analytic dependence of fractional Laplacians on metrics, leading to well-posedness results in Riemannian geometry.

Contribution

It establishes the holomorphic nature of the functional calculus for a class of operators and demonstrates real analytic dependence of fractional Laplacians on metrics.

Findings

Fractional Laplacians depend real analytically on metrics.

The functional calculus for certain operators is holomorphic.

Local well-posedness of the geodesic equation for fractional Sobolev metrics is proven.

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.