GEODESIC COMPLETENESS FOR SOBOLEV METRICS ON THE SPACE OF IMMERSED PLANE CURVES

Martins Bruveris,David Mumford,Peter W Michor

doi:10.1017/fms.2014.19

Abstract

AbstractWe study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as for initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives.

Highlights

Sobolev-type metrics on the space of plane immersed curves were independently introduced in [7, 17, 24]. They are used in computer vision, shape classification, and tracking, mainly in the form of their induced metric on shape space, which is the orbit space under the action of the reparameterization group
If the Sobolev-type metric is invariant under the reparameterization group Diff(S1), the induced metric on shape space Imm(S1, R2)/ Diff(S1) is geodesically complete
For a metric of order n 2, we extend the result to p n

Summary

Introduction

Sobolev-type metrics on the space of plane immersed curves were independently introduced in [7, 17, 24]. If G is a Sobolev-type metric of order at least 2, the Riemannian manifold (Imm(S1, R2), G) is geodesically complete. If the Sobolev-type metric is invariant under the reparameterization group Diff(S1), the induced metric on shape space Imm(S1, R2)/ Diff(S1) is geodesically complete. The latter space is an infinite-dimensional orbifold; see [17, 2.5 and 2.10]. A Sobolev metric of order 2 or higher with both a0, a1 = 0 is a geodesically incomplete metric on the space Imm(S1, R2)/ Tra of plane curves modulo translations In this case, it is possible to blow up a circle along a geodesic to infinity in finite time; see Remark 5.7. The lack of such inequalities for general M will one of the factors complicating life in higher dimensions

Background material and notation

Sobolev metrics with constant coefficients

Lower bounds on the geodesic distance

Geodesic completeness for Sobolev metrics