Heterotopic energy for Sobolev mappings

TL;DR

This paper introduces the concept of heterotopic energy for Sobolev mappings, characterizing its finiteness through homotopy on a codimension one skeleton and relating it to Sobolev and disparity energies.

Contribution

It defines heterotopic energy as a limit of Sobolev energies, establishes conditions for its finiteness, and links it to homotopy classes and disparity energy in complex manifold settings.

Findings

Heterotopic energy is finite iff mappings are homotopic on a codimension one skeleton.

Heterotopic energy equals Sobolev energy plus disparity energy when finite.

Framework applicable to non-simply connected manifolds without canonical homotopy group isomorphisms.

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.