Fractional Sobolev metrics on spaces of immersed curves

Abstract

Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves Imm(S 1 , R ) and on its Sobolev completions ℐ (S 1 , R ). We prove local well-posedness of the geodesic equations both on the Banach manifold ℐ (S 1 , R ) and on the Fréchetmanifold Imm(S 1 , R ) provided the order of the metric is greater or equal to one. In addition we show that the -metric induces a strong Riemannian metric on the Banach manifold ℐ (S 1 , R ) of the same order , provided > 3 2 . These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group.

Fractional Sobolev metrics on spaces of immersions

“…• For M = S 1 , our result specializes to the space of immersed loops in N . For loops in N = R d , 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 also to higher-dimensional and more general base manifolds M .…”

“…For curves in R n local well-posedness of the geodesic equation for integer-order metrics has been shown in [47]. This has recently been extended to fractional-order metrics in [8]. The following corollary of our main result further generalizes this to fractional-order metrics on spaces of manifold-valued curves: 5.2 Corollary.…”

Completeness and Geodesic Distance Properties for Fractional Sobolev Metrics on Spaces of Immersed Curves

“…A more detailed version of this theorem, including for results on geodesic convexity and metric completeness for curves of Sobolev regularity, is given in Theorem 4.1 . Together with the local well-posedness result for the geodesic equation [ 10 ], our result implies that the corresponding geodesic equation is globally well posed for . This was previously known only for integer-order metrics [ 14 ].…”

“…Our second main result concerns the well-posedness of the corresponding geodesic equation: Bauer–Bruveris–Kolev [ 10 ] showed that these equations are locally well posed when the order of the metric is at least 1. Here we determine the critical index for global existence, i.e, geodesic completeness of the metric:…”

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

“…The local well-posedness of the geodesic equation when the inertia operator A is a differential operator has been implicitly solved in the seminal article of Ebin and Marsden [24], see also [52,53,20,58,32,47,44,38,39], and hence for H k -metrics on diffeomorphism groups, where k is an integer. This result has been extended to invariant metrics on several related spaces of mappings, such as spaces of immersions, Riemannian metrics and the Virasoro-Bott group, see [39,6,7,3,11,4]. In a series of papers [29,28,5,41], the local and global well-posedness problem for the general EPDiff equation on Diff ∞ (T d ) or Diff H ∞ (R d ) when the inertia operator is a non-local Fourier multiplier was solved.…”