# The extension of traces for Sobolev mappings between manifolds

**Authors:** Jean Van Schaftingen  
**arXiv:** [2403.18738](https://arxiv.org/abs/2403.18738)  
**Date:** 2024-03-27  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2403.18738v1/

## Abstract

The compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$ for which the trace operator from the first-order Sobolev space of mappings $\smash{\dot{W}}^{1, p} (\mathcal{M}, \mathcal{N})$ to the fractional Sobolev-Slobodecki\u{\i} space $\smash{\smash{\dot{W}}^{1 - 1/p, p}} (\partial \mathcal{M}, \mathcal{N})$ is surjective when $1 < p < m$ are characterised. The traces are extended using a new construction which can be carried out assuming the absence of the known topological and analytical obstructions. When $p \ge m$ the same construction provides a Sobolev extension with linear estimates for maps that have a continuous extension, provided that there are no known analytical obstructions to such a control.