Surfaces of genus g ≥ 1 in 3D contact sub-Riemannian manifolds

¹ SISSA, Via Bonomea 265, 34136 Trieste, Italy
² Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Milano, Italy
³ CNRS, Laboratoire Jacques-Louis Lions, Sorbonne Université, Université de Paris, Inria, Boîte courrier 187, 75252 Paris Cedex 05 Paris, France

^* Corresponding author: eugenio.bellini@outlook.it

Received: 15 May 2023
Accepted: 2 October 2023

Abstract

We consider smooth embedded surfaces in a 3D contact sub-Riemannian manifold and the problem of the finiteness of the induced distance (i.e., the infimum of the length of horizontal curves that belong to the surface). Recently it has been proved that for a surface having the topology of a sphere embedded in a tight co-orientable structure, the distance is always finite. In this paper we study closed surfaces of genus larger than 1, proving that such surfaces can be embedded in such a way that the induced distance is finite or infinite. We then study the structural stability of the fmiteness/not-finiteness of the distance.