Lifting in compact covering spaces for fractional Sobolev mappings

Abstract

Let $\pi :\widetilde{\mathcal{𝒩}}\to \mathcal{𝒩}$ be a Riemannian covering, with $\mathcal{𝒩}$, $\widetilde{\mathcal{𝒩}}$ smooth compact connected Riemannian manifolds. If $\mathcal{ℳ}$ is an $m$-dimensional compact simply connected Riemannian manifold, and , we prove that every mapping $u\in W^{s,p}\left(\mathcal{ℳ},\mathcal{𝒩}\right)$ has a lifting in $W^{s,p}$; i.e., we have $u=\pi \circ ũ$ for some mapping $ũ\in W^{s,p}\left(\mathcal{ℳ},\widetilde{\mathcal{𝒩}}\right)$. Combined with previous contributions of Bourgain, Brezis and Mironescu and Bethuel and Chiron, our result settles completely the question of the lifting in Sobolev spaces over covering spaces.

The proof relies on an a priori estimate of the oscillations of $W^{s,p}$ maps with and , in dimension $1$. Our argument also leads to the existence of a lifting when and , provided there is no topological obstruction on $u$; i.e., $u=\pi \circ ũ$ holds in this range provided $u$ is in the strong closure of $C^{\infty }\left(\mathcal{ℳ},\mathcal{𝒩}\right)$.

However, when , $sp=1$ and $m\ge 2$, we show that an (analytical) obstruction still arises, even in the absence of topological obstructions. More specifically, we construct some map $u\in W^{s,p}\left(\mathcal{ℳ},\mathcal{𝒩}\right)$ in the strong closure of $C^{\infty }\left(\mathcal{ℳ},\mathcal{𝒩}\right)$ such that $u=\pi \circ ũ$ does not hold for any $ũ\in W^{s,p}\left(\mathcal{ℳ},\widetilde{\mathcal{𝒩}}\right)$.