Sobolev homeomorphisms are dense in volume preserving automorphisms

Abstract

In this paper we prove a weak version of Lusin's theorem for the space of Sobolev-(1,p) volume preserving homeomorphisms on closed and connected n-dimensional manifolds, n≥3, for p<n−1. We also prove that if p>n this result is not true. More precisely, we obtain the density of Sobolev-(1,p) homeomorphisms in the space of volume preserving automorphisms, for the weak topology. Furthermore, the regularization of an automorphism in a uniform ball centered at the identity can be done in a Sobolev-(1,p) ball with the same radius centered at the identity.