Fractional Sobolev metrics on spaces of immersions

TL;DR

This paper proves local well-posedness of geodesic equations for fractional Sobolev metrics on spaces of immersions and diffeomorphisms, extending previous results to a broad class of variational PDEs.

Contribution

It establishes local well-posedness for fractional Sobolev metrics of order one and higher, leveraging real analytic dependence of fractional Laplacians on Riemannian metrics.

Findings

Well-posedness of geodesic equations for fractional Sobolev metrics.

Extension of results to manifold-valued curves and surfaces.

Applicability to Euler-Arnold equations and related PDEs.

Abstract

We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic dependence of fractional Laplacians on the underlying Riemannian metric. It extends several previous results and applies to a wide range of variational partial differential equations, including the well-known Euler-Arnold equations on diffeomorphism groups as well as the geodesic equations on spaces of manifold-valued curves and surfaces.