Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 311-335

SOME REMARKS ON CONTACT VARIATIONS IN THE FIRST HEISENBERG GROUP

Sebastiano Golo

University of Trento, Department of Mathematics
Via Sommarive 14, 38123 Povo, Italy; s.nicolussigolo 'at' unitn.it

Abstract. We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, we show that if f : R² → R is a C¹-intrinsic function, and ∇^f∇^ff = 0, then the first contact variation of the sub-Riemannian area of its intrinsic graph is zero and the second contact variation is positive.

2010 Mathematics Subject Classification: Primary 53C17, 49Q20.

Key words: Sub-Riemannian geometry, sub-Riemannian perimeter, Heisenberg group, Bernstein's problem, contact diffeomorphisms.

Reference to this article: S. Golo: Some remarks on contact variations in the first Heisenberg group. Ann. Acad. Sci. Fenn. Math. 43 (2018), 311-335.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4312