An Obata-type Theorem in CR Geometry

A Lichnerowicz-type result on a seven-dimensional quaternionic contact manifold

“…More precisely, as a consequence of his more general result, it can be stated, that the equality in (1.2) is attained iff the Riemannian manifold (M, g) is isometrical to the unit sphere S n (1) endowed with the round metric, as (1.1) holds. This result has provoked a similar question in the sub-Riemannian geometry and in particular in the CR geometry, where the problem is successfully solved, see [28,29,35].…”

The Obata first eigenvalue theorems on a seven dimensional quaternionic contact manifold

“…As far as the Obata type result is concerned, the most general result valid on a complete CR manifold was proven in [21] under the assumption of a divergence-free pseudohermitian torsion. In the compact case, [26,27,22] proved the Obata type theorem on a compact strictly pseudoconvex pseudohemitian manifold which satisfies the Lichnerowicz-type bound. In dimension three, in all of the above results it is assumed that the CR-Paneitz operator is non-negative.…”

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

“…Following [34] the above cited results on a compact CR manifold focused on adding a corresponding inequality for n = 1, 2 or characterizing the equality case mainly in the vanishing Webster-torsion case (the Sasakian case). The general case on a compact CR manifold satisfying the Lichnerowicz type condition was proved in [60,61] using the results and the method of [45] while introducing a new integration by parts step proving the vanishing of the Webster torsion assuming the first eigenvalue is equal to n (for the three dimensional case see [46]). On the other hand, a generalization of the Obata result in the complete non-compact case was achieved in [45], where the standard Sasakian structure on the unit sphere was characterized through the existence of a non-trivial solution of a (horizontal) Hessian equation on a complete with respect to the associated Riemannian metric CR manifold with a divergence free Webster torsion.…”