Finite Distortion Sobolev Mappings between Manifolds are Continuous

Abstract

AbstractWe prove that if $M$ and $N$ are Riemannian, oriented $n$-dimensional manifolds without boundary and additionally $N$ is compact, then Sobolev mappings in $W^{1,n}(M,N)$ of finite distortion are continuous. In particular, $W^{1,n}(M,N)$ mappings with almost everywhere positive Jacobian are continuous. This result has been known since 1976 in the case of mappings in $W^{1,n}(\Omega,{\mathbb R}^n)$, where $\Omega\subset{\mathbb R}^n$ is an open set. The case of mappings between manifolds is much more difficult.