Beltrami system and 1-quasiconformal embeddings in higher dimensions

TL;DR

This paper establishes conditions for 1-quasiconformal parameterizations of surfaces in higher dimensions, revealing limitations of classical rigidity theorems when embedding into spaces of dimension greater than the domain.

Contribution

It derives necessary and sufficient conditions for 1-quasiconformal embeddings in higher dimensions and demonstrates the failure of Liouville's rigidity in these cases.

Findings

Conditions for 1-quasiconformal parameterizations in R^{n+1}

Application to specific hypersurfaces like cylinders and ellipsoids

Liouville theorem does not extend to higher-dimensional embeddings

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.