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Heterotopic energy for Sobolev mappings

Antoine Detaille, Jean Van Schaftingen

Abstract

We study the notion of heterotopic energy defined as the limit of Sobolev energies of Sobolev mappings in a given homotopy class approximating almost everywhere a given Sobolev mapping. We show that the heterotopic energy is finite if and only if the mappings in the corresponding homotopy classes are homotopic on a codimension one skeleton of a triangulation of the domain. When this is the case, the heterotopic energy of a mapping is the sum of its Sobolev energy and its disparity energy, defined as the minimum energy of a bubble to pass between these homotopy classes. At the more technical level, we rely on a framework that works when the target and domain manifolds are not simply connected and there is no canonical isomorphism between homotopy groups with different basepoints.

Heterotopic energy for Sobolev mappings

Abstract

We study the notion of heterotopic energy defined as the limit of Sobolev energies of Sobolev mappings in a given homotopy class approximating almost everywhere a given Sobolev mapping. We show that the heterotopic energy is finite if and only if the mappings in the corresponding homotopy classes are homotopic on a codimension one skeleton of a triangulation of the domain. When this is the case, the heterotopic energy of a mapping is the sum of its Sobolev energy and its disparity energy, defined as the minimum energy of a bubble to pass between these homotopy classes. At the more technical level, we rely on a framework that works when the target and domain manifolds are not simply connected and there is no canonical isomorphism between homotopy groups with different basepoints.

Paper Structure

This paper contains 10 sections, 48 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. The heterotopic energy on Sobolev spaces
  3. Continuous, VMO, and Sobolev homotopies
  4. Heterotopic energy
  5. Finiteness criterion
  6. Improved upper estimate
  7. Topological disparities
  8. The topological energy
  9. The disparity energy
  10. Bubbling

Key Result

Theorem 1.1

For every $u, v \in \mathrm{C}^\infty (\mathcal{M}, \mathcal{N})$, In particular, $\mathfrak{E}^{1, m}_{\mathrm{het}} (u, v) < \infty$ if and only if $u$ and $v$ are homotopic on an $(m-1)$-dimensional triangulation of $\mathcal{M}$.

Theorems & Definitions (90)

  • Theorem 1.1
  • Corollary 1.2: Mappings into spheres
  • Corollary 1.3: Mappings into projective spaces
  • Corollary 1.4
  • Theorem 1.5
  • Proposition 2.1: Brezis & Nirenberg Brezis_Nirenberg_1995*Lemma A.19
  • Proposition 2.2: Brezis & Nirenberg Brezis_Nirenberg_1995*Lemma A.20
  • Proposition 2.3: Brezis & Nirenberg Brezis_Nirenberg_1996*Corollary 3
  • Proposition 2.4
  • Proposition 2.5
  • ...and 80 more