Calculus of Variations and Geometric Measure Theory

V. Magnani - E. Stepanov - D. Trevisan

A rough calculus approach to level sets in the Heisenberg group

created by magnani on 27 Oct 2016
modified on 08 Nov 2018


Published Paper

Inserted: 27 oct 2016
Last Updated: 8 nov 2018

Journal: J. Lond. Math. Soc.
Volume: 97
Number: 3
Pages: 495–522
Year: 2018

ArXiv: 1610.08873 PDF


We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in $\mathbb{R}^2$. These equations can be seen as a sub-Riemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they enjoy all the natural well-posedness properties, thus allowing for a "good calculus" on nonsmooth level sets. We apply these results to prove an area formula for the intrinsic measure of level sets, along with the corresponding coarea formula.