Topological singular set of vector-valued maps, I: Applications to manifold-constrained Sobolev and BV spaces

TL;DR

This paper introduces a topological operator for vector-valued maps into manifolds, capturing essential topological features and characterizing limits of smooth maps in Sobolev spaces, with applications to BV maps.

Contribution

It defines a new operator $ extbf{S}$ that generalizes the distributional Jacobian for manifold-valued maps and characterizes their strong limits in Sobolev spaces.

Findings

Operator $ extbf{S}$ captures topological information of maps.

Characterizes strong limits of smooth maps in Sobolev spaces.

Applications to BV maps and future work on Ginzburg-Landau functionals.

Abstract

We introduce an operator $\mathbf{S}$ on vector-valued maps $u$ which has the ability to capture the relevant topological information carried by $u$. In particular, this operator is defined on maps that take values in a closed submanifold $N$ of the Euclidean space $\mathbb{R}^m$, and coincides with the distributional Jacobian in case $N$ is a sphere. The range of $\mathbf{S}$ is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use $\mathbf{S}$ to characterise strong limits of smooth, $N$-valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivi\`ere. We also discuss applications to the study of manifold-valued maps of bounded variation. In a companion paper, we will consider applications to the asymptotic behaviour of minimisers of Ginzburg-Landau type functionals.