Singular extension of critical Sobolev mappings with values into complete Riemannian manifolds

TL;DR

This paper extends singular Sobolev map extension results to a broader class of Riemannian manifolds with positive reach, using a criterion by Petrunin, and demonstrates the results with examples.

Contribution

It generalizes previous extension theorems for Sobolev maps to include manifolds with positive reach, broadening applicability.

Findings

Extended maps satisfy exponential weak-type Sobolev-Marcinkiewicz estimates.

Applicable to manifolds with positive reach, including certain hyperbolic and Euclidean spaces.

Provides illustrative examples demonstrating the theory.

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 Poincar\'{e} 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.