Nonexistence of horizontal Sobolev surfaces in the Heisenberg group

TL;DR

This paper proves that the involutivity condition remains necessary for the integrability of Sobolev surfaces within the horizontal distribution of the Heisenberg group, addressing a previously open question.

Contribution

It extends the classical involutivity condition for integrability to Sobolev surfaces in the Heisenberg group, resolving an open problem.

Findings

Involutivity is necessary for Sobolev surface integrability in the Heisenberg group

The study confirms the necessity condition in a non-smooth setting

Answers a specific open question in geometric analysis

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.