Topological obstructions to continuity of Orlicz-Sobolev mappings of finite distortion

TL;DR

This paper studies when mappings of finite distortion between Riemannian manifolds are continuous, revealing that certain topological properties of the target manifold determine the continuity of these mappings.

Contribution

It establishes topological conditions under which Orlicz-Sobolev mappings of finite distortion are continuous, and constructs counterexamples showing the necessity of these conditions.

Findings

Mappings with $Df otin L^n$ can be discontinuous.

Topological properties of the target manifold influence mapping continuity.

Continuity depends on the universal cover's topology, not just local properties.

Abstract

In the paper we investigate continuity of Orlicz-Sobolev mappings $W^{1,P}(M,N)$ of finite distortion between smooth Riemannian $n$-manifolds, $n\geq 2$, under the assumption that the Young function $P$ satisfies the so called divergence condition $\int_1^\infty P(t)/t^{n+1}\, dt=\infty$. We prove that if the manifolds are oriented, $N$ is compact, and the universal cover of $N$ is not a rational homology sphere, then such mappings are continuous. That includes mappings with $Df\in L^n$ and, more generally, mappings with $Df\in L^n\log^{-1}L$. On the other hand, if the space $W^{1,P}$ is larger than $W^{1,n}$ (for example if $Df\in L^n\log^{-1}L$), and the universal cover of $N$ is homeomorphic to $\mathbb{S}^n$, $n\neq 4$, or is diffeomorphic to $\mathbb{S}^n$, $n=4$, then we construct an example of a mapping in $W^{1,P}(M,N)$ that has finite distortion and is discontinuous. This demonstrates a new global-to-local phenomenon: both finite distortion and continuity are local properties, but a seemingly local fact that finite distortion implies continuity is a consequence of a global topological property of the target manifold $N$.