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Completeness of reparametrization-invariant Sobolev metrics on the space of surfaces

Martin Bauer, Cy Maor, Benedikt Wirth

TL;DR

This work establishes metric and geodesic completeness for reparametrization-invariant Sobolev metrics on spaces of immersed surfaces, extending prior completeness results from curves to 2D submanifolds. By formulating an abstract completeness criterion for open subsets of complete manifolds and leveraging Michael–Simon–Sobolev inequalities, the authors prove completeness and minimizing geodesic existence for a curvature-weighted $H^k$ metric on immersed surfaces and for the induced shape space. They further show the quotient shape space is a complete metric space with optimal reparametrizations and that minimizing geodesics exist both on the immersion space and in the shape space. The curvature-weighted $H^k$ metric is explicitly constructed to satisfy the required bounds, enabling a robust, general framework for completeness in infinite-dimensional spaces of submanifolds with potential extensions to broader geometric settings.

Abstract

We study reparametrization-invariant Sobolev-type Riemannian metrics on the space of immersed surfaces and establish conditions ensuring metric and geodesic completeness as well as the existence of minimizing geodesics. This provides the first extension of completeness results for immersed curves, originating from works of Bruveris, Michor, and Mumford, and validates an earlier conjecture of Mumford on completeness properties of general spaces of immersions in this important case. The result is obtained by recasting earlier approaches to completeness on manifolds of mappings as a general completeness criterion for infinite-dimensional Riemannian manifolds that are open subsets of a complete Riemannian manifold and by combining it with geometric estimates based on the Michael--Simon--Sobolev inequality to establish the completeness for specific Sobolev metrics on immersed surfaces.

Completeness of reparametrization-invariant Sobolev metrics on the space of surfaces

TL;DR

This work establishes metric and geodesic completeness for reparametrization-invariant Sobolev metrics on spaces of immersed surfaces, extending prior completeness results from curves to 2D submanifolds. By formulating an abstract completeness criterion for open subsets of complete manifolds and leveraging Michael–Simon–Sobolev inequalities, the authors prove completeness and minimizing geodesic existence for a curvature-weighted metric on immersed surfaces and for the induced shape space. They further show the quotient shape space is a complete metric space with optimal reparametrizations and that minimizing geodesics exist both on the immersion space and in the shape space. The curvature-weighted metric is explicitly constructed to satisfy the required bounds, enabling a robust, general framework for completeness in infinite-dimensional spaces of submanifolds with potential extensions to broader geometric settings.

Abstract

We study reparametrization-invariant Sobolev-type Riemannian metrics on the space of immersed surfaces and establish conditions ensuring metric and geodesic completeness as well as the existence of minimizing geodesics. This provides the first extension of completeness results for immersed curves, originating from works of Bruveris, Michor, and Mumford, and validates an earlier conjecture of Mumford on completeness properties of general spaces of immersions in this important case. The result is obtained by recasting earlier approaches to completeness on manifolds of mappings as a general completeness criterion for infinite-dimensional Riemannian manifolds that are open subsets of a complete Riemannian manifold and by combining it with geometric estimates based on the Michael--Simon--Sobolev inequality to establish the completeness for specific Sobolev metrics on immersed surfaces.

Paper Structure

This paper contains 20 sections, 23 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Contributions of the article
  3. Open Questions and Future Work
  4. Structure of the Article
  5. Completeness for Strong Riemannian Metrics
  6. Spaces of Immersions and Notation
  7. Sobolev Spaces
  8. The Manifold of Immersions
  9. The Group of Diffeomorphisms and the Shape Space of Unparametrized Surfaces
  10. The Induced Geometry of a Surface $f$
  11. Sobolev (Semi-)Norms
  12. Riemannian Metrics on Spaces of Immersions
  13. Intrinsic Sobolev Embedding Theorems
  14. Bounding Geometric Quantities on Metric Balls
  15. Completeness for abstract Riemannian metrics on the Space of Surfaces
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $M$ be a two-dimensional, closed manifold, let $d,l\geq 3$ and let $\mathrm{Imm}^l(M,\mathbb R^d)$ be the space of all immersions of $M$ into $\mathbb R^d$ of regularity $H^l$, equipped with the Riemannian metric $G^k$ from eq:curvatureweightedmetricIntro of order $3\leq k\leq l$. We have:

Theorems & Definitions (54)

  • Theorem 1.1: Completeness Properties of the Space of Immersed Surfaces
  • Conjecture : Completeness Properties on the Space of Immersed Submanifolds
  • Theorem 2.1: Inheritance of Completeness Properties
  • Remark 2.2: Metric Completeness Implies Geodesic Completeness
  • proof
  • Remark 2.3: Smoothness Requirements
  • Example 2.4: $\ell_2$-Sequences Bounded Away from Zero
  • Remark 3.1: Fiber Metric for Elementary Tensors
  • Remark 3.2: Fiber Metric for Bundle Homomorphisms
  • Lemma 3.3: Sobolev regularity of derivatives
  • ...and 44 more