Finite Distortion Sobolev Mappings between Manifolds are Continuous

We prove that if M and N are Riemannian, oriented n-dimensional manifolds without boundary and additionally N is compact, then Sobolev mappings W1,n(M,N) of finite distortion are continuous. In particular, W1,n(M,N) mappings with almost everywhere positive Jacobian are continuous. This result has been known since 1976 in the case of mappings W1,n(Ω,ℝn), where Ω⊂ℝn is an open set. The case of mappings between manifolds is much more difficult.