 
Summary: Stability of isometric maps in the Heisenberg group
Nicola Arcozzi
and Daniele Morbidelli
13th September 2005
Abstract
In this paper we prove some approximation results for biLipschitz maps in
the Heisenberg group. Namely, we show that a biLipschitz map with biLipschitz
constant close to one can be pointwise approximated, quantitatively on any fixed
ball, by an isometry. This leads to an approximation in BMO norm for the map's
Pansu derivative. We also prove that a global quasigeodesic can be approximated
by a geodesic on any fixed segment.
1 Introduction
In 1961 Fritz John proved the following stability estimates. Let f : Rn
Rn
be a
biLipschitz map such that f(0) = 0 and the Lipschitz constant of f and f1
is less
than 1 + . Then for any ball B = B(0, R), there is T O(n) such that
f(x)  Tx CnR, x B and (1.1)
1
