Nonexistence of horizontal Sobolev surfaces in the Heisenberg group

Abstract

Involutivity is a well known necessary condition for integrability of smooth tangent distributions. We show that this condition is still necessary for integrability with Sobolev surfaces. We specialize our study to the left invariant horizontal distribution of the first Heisenberg group $\H^1$. Here we answer a question raised in a paper by Z.M.Balogh, R.Hoefer-Isenegger, J.T.Tyson.