Singular extension of critical Sobolev mappings with values into complete Riemannian manifolds
Federico Luigi Dipasquale
TL;DR
The paper extends the singular extension results for critical Sobolev mappings with values into a compact target to complete, non-compact Riemannian targets that admit an isometric Euclidean embedding with positive reach (tubed embeddings). It leverages Petrunin's criterion to identify such targets and adapts the Bulanyi–Van Schaftingen averaging-extension construction to obtain a trace-preserving extension $U$ with a weak-type exponential Sobolev-Marcinkiewicz estimate in $W^{1,1}$, including a refined bound when the trace is bounded. The main contributions are Theorems I–III in this non-compact setting, explicit energy bounds, and a collection of illustrative examples (e.g., warped products, universal covers, conical/ALE manifolds) that lie beyond the compact‑target regime. This work bridges nonlinear extension theory with geometric embedding criteria and sets the stage for extensions of linear-type estimates (à la VS) to broader noncompact targets.
Abstract
Relying on a recent criterion, due to A.~Petrunin [18], to check if a complete, non-compact, Riemannian manifold admits an isometric embedding into a Euclidean space with positive reach, we extend to manifolds with such property the singular extension results of B.~Bulanyi and J.~Van~Schaftingen [5] for maps in the critical, nonlinear Sobolev space $W^{m/(m+1),m+1}\left(X^m,\mathcal{N}\right)$, where $m \in \mathbb{N} \setminus \{0\}$, $\mathcal{N}$ is a compact Riemannian manifold, and $X^m$ is either the sphere $\mathbb{S}^m = \partial \mathbb{B}^{m+1}_+$, the plane $\mathbb{R}^m$, or again $\mathbb{S}^m$ but seen as the boundary sphere of the Poincaré ball model of the hyperbolic space $\mathbb{H}^{m+1}$. As in [5], we obtain that the extended maps satisfy an exponential weak-type Sobolev-Marcinkiewicz estimate. Finally, we provide some illustrative examples.