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.
Highlights
We thank Martins Bruveris, Boris Kolev, Andreas Kriegl, Peer Kunstmann, and Lutz Weis for helpful discussions
Our generalization to fractional-order metrics builds on recent results about the smoothness of the functional calculus of sectorial operators [6]
For M = N our result specializes to the diffeomorphism group Diff(M), which is an open subset of Imm(M, M)
Summary
This follows from Theorems 4.4 and 4.6. The result unifies and extends several previously known results:. For integer-order metrics, local well-posedness on the space of immersions from M to N has been shown in [11]. In the absence of global coordinate systems for these mapping spaces, we recast the geodesic equation using an auxiliary covariant derivative following [11]; see Lemma 2.6 and Theorem 4.3. On Diff(M) we obtain local well-posedness of the geodesic equation for Sobolev metrics of order p ∈ [1/2, ∞); see Corollary 5.1. For loops in N = Rd , local well-posedness has been shown by different methods (reparameterization to arc length) in [8]. Our analysis extends this result to manifold-valued loops and to higher-dimensional and more general base manifolds M
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