Fractional Sobolev metrics on spaces of immersions

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.

Fractional Sobolev metrics on spaces of immersed curves

“…From here the result follows using the definition of the inertia operator A c , the product rule for the term ∇ ∂s (Ψ c (c t , c t )v), by using the formula Proof of Theorem 3.8. For closed curves, i.e., D = S 1 , this result can be found in [10,Theorem 4.4], see also [28,6]. In the following we will focus on the case of open curves, where the proof will be slightly more involved due to the existence of a boundary.…”

“…Proof. The first part of this result can be found in [10,Theorem 2.4], while the second part follows directly from the definition of the space…”

“…The local wellposedness results as summarized in the following theorem are based on the seminal method of Ebin and Marsden [20]. They are known in the case of closed curves, see [10,28,6], but to the best of our knowledge they are new for the case of open curves. However, as local well-posedness is not the focus of the current article, we postpone the proof of this result to the Appendix A.2.…”

Well-posedness of the EPDiff equation with a pseudo-differential inertia operator

“…In this article we extend this analysis to the EPDiff equation on compact manifolds, which requires us to deal with inertia operators which are general Pseudo Differential operators. Simultaneously to this article, the first author and collaborators proved in [8] local well-posedness of geodesic equations for fractional order metrics on the space of immersions of a manifold M with values in another manifold N . The class of operators studied in [8] is defined via holomorphic functional calculus of the Laplace operator.…”

“…Simultaneously to this article, the first author and collaborators proved in [8] local well-posedness of geodesic equations for fractional order metrics on the space of immersions of a manifold M with values in another manifold N . The class of operators studied in [8] is defined via holomorphic functional calculus of the Laplace operator. In the special case of M being N their results agree with the first part of the main theorem of the present article (the local well-posedness of the geodesic equations), albeit for a different class of inertia operators and using a different method of proof (our strategy is heavily based on the group structure of D q (M ) and is valid for general abstract pseudo differential operators, see the comments below).…”