 
Summary: THE HESSIAN OF THE DISTANCE FROM A SURFACE IN THE
HEISENBERG GROUP
NICOLA ARCOZZI, FAUSTO FERRARI
Abstract. Given a smooth surface S in the Heisenberg group, we compute the Hessian of the
function measuring the CarnotCharath´eodory distance from S in terms of the Mean Curvature
of S and of an "imaginary curvature" which was introduced in [2] in order to find the geodesics
which are metrically normal to S. Explicit formulae are given when S is a plane or the metric
sphere.
Contents
1. Introduction 1
2. Notation and preliminaries 4
3. The Hessian of the function measuring the distance from a smooth surface 8
4. Mean Curvature 14
5. Examples 18
5.1. Nonvertical planes 18
5.2. The CarnotCharath´eodory sphere and the Hessian of the distance function 20
References 22
1. Introduction
In this article we continue to study the properties of the function "signed distance from a
surface S", S, in the Heisenberg group, started in [2]. In particular we are interested in the
