Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics

Abstract

We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields in R n . It turns out that in many cases all such maps can be obtained as compositions of suitable dilations, inversions and isometries. Our methods involve a study of the singular Riemannian metric associated with the vector fields. In particular, we identify some conformally invariant cones related to the Weyl tensor. The knowledge of such cones enables us to classify all umbilical hypersurfaces.

Quasiconformal mappings on the Grushin plane

“…This a rigidity theorem similar to (and deriving from) the well-known classification of conformal homeomorphisms of the Euclidean plane. A similar rigidity result, using more sophisticated arguments, is found in Morbidelli [10] for maps between domains in certain higher-dimensional Grushin spaces, though the case of the Grushin plane is not explicitly addressed there. 0) be an orientation-preserving homeomorphism which is metrically conformal.…”

The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven

“…A closely related property is the so called rigidity property of quasiconformal or multicontact maps, also referred to as Liouville's property, but where the question is the finite dimensionality of the group of (locally defined) quasiconformal or multicontact maps, see [80], [74], [23], [68], [69], [65], [24], [56].…”

The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds

“…A closely related property is the so called rigidity property of quasiconformal or multicontact maps, also referred to as Liouville's property, but where the question is the finite dimensionality of the group of (locally defined) quasiconformal or multicontact maps, see [230], [199], [68], [192], [193], [185], [77], [163].…”