Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sobolev metrics on spaces of manifold valued curves

Martin Bauer, Cy Maor, Peter W. Michor

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).

Sobolev metrics on spaces of manifold valued curves

Abstract

We study completeness properties of reparametrization invariant Sobolev metrics of order 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).

Paper Structure

This paper contains 27 sections, 30 theorems, 211 equations.

Table of Contents

  1. Introduction and main results
  2. Background
  3. Main Result
  4. Further contributions of the article
  5. Main ideas in the proof and structure of the article
  6. Local well-posedness
  7. Metric and geodesic completeness
  8. Existence of minimal geodesics
  9. Spaces of manifold valued functions and immersions
  10. Reparametrization invariant Sobolev metrics on spaces of curves
  11. The metric and geodesic equation in the smooth category
  12. The induced metric on Sobolev immersions
  13. Local well-posedness of the geodesic equation
  14. Estimates
  15. Metric and geodesic completeness
  16. ...and 12 more sections

Key Result

Theorem 1

Let $D=[0,2\pi]$ or $D=S^1$, and let $G$ be the scale invariant Sobolev metric def:SobMetric of order $n\geq 2$. The following completeness properties hold: For $D=S^1$ the results continue to hold for the family of constant coefficient Sobolev metrics.

Theorems & Definitions (61)

  • Theorem : Main Theorem
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 51 more