Beltrami system and 1-quasiconformal embeddings in higher dimensions

Abstract

In this paper we derive necessary and sufficient conditions for a smooth surface in Rn+1 to admit a local 1-quasiconformal parameterization by a domain in Rn (n >= 3). We then apply these conditions to specific hypersurfaces such as cylinders, paraboloids, and ellipsoids. As a consequence, we show that the classical Liouville theorem about the rigidity of 1-quasiconformal maps between domains in Rn with n >= 3 does not extend to embeddings of domains into a higher dimensional space.