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We report on some aspects and recent progress in certain problems in thesub-Riemannian CR and quaternionic contact (QC) geometries. The focus are thecorresponding Yamabe problems on the round spheres, the Lichnerowicz-Obatafirst eigenvalue estimates, and the relation between these two problems. Amotivation from the Riemannian case highlights new and old ideas which are thendeveloped in the settings of Iwasawa sub-Riemannian geometries.

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