An improved dense class in Sobolev spaces to manifolds

TL;DR

This paper advances the strong density theory for manifold-valued Sobolev maps by proving density of an improved uncrossed class $\mathscr{R}^{\textnormal{uncr}}_{m-\lfloor sp\rfloor-1}$ in $W^{s,p}(Q^{m};\mathscr{N})$ for $sp<m$, addressing a question of Brezis–Mironescu. It starts by extending the singular projection method to the full applicability range for targets with $(\ell-2)$-connectedness and, when possible, provides density for the uncrossed class via a projection-based approach. To handle general target manifolds where projection cannot guarantee uncrossed singularities, the authors develop a two-stage strategy: a topological crossing-removal procedure that uncrosses the singular set, followed by a shrinking/energy-control step to preserve Sobolev regularity. The result is a robust framework that yields strong density results for a broad class of manifold targets in fractional Sobolev spaces and resolves longstanding questions about the necessity of crossing-free singular sets in density proofs. Collectively, the work extends techniques from Federer–Fleming-type projection methods and nonlinear topological deformations to fractional orders, with implications for the approximation of geometric variational problems into manifolds.

Abstract

We consider the strong density problem in the Sobolev space $ W^{s,p}(Q^{m};\mathscr{N}) $ of maps with values into a compact Riemannian manifold $ \mathscr{N} $. It is known, from the seminal work of Bethuel, that such maps may always be strongly approximated by $ \mathscr{N} $-valued maps that are smooth outside of a finite union of $ (m -\lfloor sp \rfloor - 1) $-planes. Our main result establishes the strong density in $ W^{s,p}(Q^{m};\mathscr{N}) $ of an improved version of the class introduced by Bethuel, where the maps have a singular set without crossings. This answers a question raised by Brezis and Mironescu. In the special case where $ \mathscr{N} $ has a sufficiently simple topology and for some values of $ s $ and $ p $, this result was known to follow from the method of projection, which takes its roots in the work of Federer and Fleming. As a first result, we implement this method in the full range of $ s $ and $ p $ in which it was expected to be applicable. In the case of a general target manifold, we devise a topological argument that allows to remove the self-intersections in the singular set of the maps obtained via Bethuel's technique.

Figures (8)

• Figure 1: Skeletons and their dual skeletons
• Figure 2: The retraction $g$ and its singular set
• Figure 3: Planar view of the faces of $Q^{3}$ without the top one
• Figure 4: The well $W_{\mu}$, with singularities before and after uncrossing
• Figure 5: Uncrossing plane singularities in one direction

Theorems & Definitions (31)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Definition 1.5
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• proof : Proof of Lemma \ref{['lemma:necessary_condition_projection']}
• proof : Proof of Lemma \ref{['lemma:existence_of_projection']}