Stability of isometric maps in the Heisenberg group

Abstract

In this paper we prove 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 in 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.