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

TL;DR

This paper explores eigenvalue estimates and Yamabe problems in CR and quaternionic contact geometries, extending classical Riemannian results to sub-Riemannian settings with recent progress and new insights.

Contribution

It advances understanding of eigenvalue bounds and Yamabe problem solutions in CR and quaternionic contact geometries, connecting these to classical Riemannian results.

Findings

Derived new eigenvalue estimates in CR and QC geometries

Established relations between Yamabe problems and eigenvalues in these settings

Extended classical theorems to sub-Riemannian geometries

Abstract

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.