Continuity of weakly differentiable mappings of finite distortion

There are several definitions of mappings of finite distortion, which in essence reduce to the following: a weakly differentiable mapping f:\Omega\subset R^n->R^n has finite distortion if at almost all points either its Jacobian determinant J_f is positive, or its derivative Df vanishes. 

 

Vodop'yanov and Gol'dshtein proved in 1976 that every finite distortion mapping in W^{1,n}(\Omega,R^n) has a continuous representative. This result has been extended later (Iwaniec, Martin, Koskela, Onninen) to spaces slightly larger than W^{1,n}, in particular to mappings with Df in L^n Log^{-1}, with more or less the same proof.

 

However, the proof essentially used the fact that the target space - R^n - could be retracted onto an n-dimensional ball and thus did not generalize to mappings between closed manifolds. 

 

Recently, together  with Piotr Hajlasz and M. Reza Pakzad, we proved 

 

Theorem: If M,N are oriented, closed n-manifolds, then any mapping of finite distortion (in particular - a mapping with positive Jacobian a.e.) in W^{1,n}(M,N) is continuous.

 

This result has direct application to the regularity of Sobolev immersions f:\Omega->R^n  (paper is in preparation).

 

Clearly, one should ask whether - as in the case of Euclidean domains - the result generalizes to Orlicz-Sobolev spaces (as the space of functions with Df in L^n Log^{-1}). It should be expected, since both finite distortion and continuity are essentially local properties of a mapping. 

 

Question: Assume M,N are oriented, closed n-manifolds. Is every mapping with Df in L^n Log^{-1} (M,N) continuous?

 

It turns out that the answer depends deeply on the topology of the target manifold: 

If the universal cover of N has deRham cohomology different that that of S^n, then the answer is YES. However, if the universal cover of N is an n-sphere, the answer is NO. Thus there is a surprising global obstruction to the standard "local-to-global" reasoning.

 

The latter part is joint work with Piotr Hajlasz.