Sobolev metrics on spaces of manifold valued curves

Abstract

We study completeness properties of reparametrization invariant Sobolev metrics of order $n\ge 2$ on the space of manifold valued open and closed immersed curves. In particular, for several important cases of metrics, we show that Sobolev immersions are metrically and geodesically complete (thus the geodesic equation is globally well-posed). These results were previously known only for closed curves with values in Euclidean space. For the class of constant coefficient Sobolev metrics on open curves, we show that they are metrically incomplete, and that this incompleteness only arises from curves that vanish completely (unlike "local" failures that occur in lower order metrics).

arXiv commentv2: minor changes

References in corpus (11)

• An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
• Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane Curves
• Shape Analysis on Lie Groups with Applications in Computer Animation
• Analyzing Dynamical Brain Functional Connectivity As Trajectories on Space of Covariance Matrices
• Geodesic Equations on Diffeomorphism Groups
• Manifolds of mappings for continuum mechanics
• Sobolev Metrics on Diffeomorphism Groups and the Derived Geometry of Spaces of Submanifolds
• Local well-posedness of the EPDiff equation: a survey
• Fractional Sobolev metrics on spaces of immersions
• Metrics on Spaces of Immersions where Horizontality Equals Normality
• Completeness of Length-Weighted Sobolev Metrics on the Space of Curves

Cited by in corpus (2)

• A new variational model for shape graph registration with partial matching constraints
• Brownian motion on spaces of discrete regular curves