  We report on some aspects and recent progress in certain problems in the
sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the
corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata
first eigenvalue estimates, and the relation between these two problems. A
motivation from the Riemannian case highlights new and old ideas which are then
developed in the settings of Iwasawa sub-Riemannian geometries.

Differential Geometry;Analysis of PDEs