On the local character of the extension of traces for Sobolev mappings

TL;DR

This paper characterizes when a map between compact Riemannian manifolds can be extended to a Sobolev map, showing that extension obstructions are purely local in nature.

Contribution

It establishes a local criterion for the extension of Sobolev mappings between compact Riemannian manifolds, clarifying the nature of extension obstructions.

Findings

Extension of Sobolev maps depends on local conditions.

Global obstructions are characterized as local phenomena.

Provides a criterion for Sobolev trace extension on manifolds.

Abstract

We prove that a mapping $u \colon \mathcal{M}'\to \mathcal{N}$, where $\mathcal{M}'$ and $ \mathcal{N}$ are compact Riemannian manifolds, is the trace of a Sobolev mapping $U \colon \mathcal{M}' \times [0, 1) \to \mathcal{N}$ if and only if it is on some open covering of $\mathcal{M}'$. In the global case where $\mathcal{M}$ is a compact Riemannian manifold with boundary, this implies that the analytical obstructions to the extension of a mapping $u \colon \partial \mathcal{M}\to \mathcal{N}$ to some Sobolev mapping $U \colon \mathcal{M} \to \mathcal{N}$ are purely local.