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

Abstract

We show that a compact quaternionic contact manifold of dimension seven that satisfies a Lichnerowicz-type lower Ricci-type bound and has the $P$-function of any eigenfunction of the sub-Laplacian non-negative achieves its smallest possible eigenvalue only if the structure is qc-Einstein. In particular, under the stated conditions, the lowest eigenvalue is achieved if and only if the manifold is qc-equivalent to the standard $3$-Sasakian sphere.

References in corpus (4)

• Quaternionic contact Einstein structures and the quaternionic contact Yamabe problem
• A new characterization of the CR shpere and the sharp eigenvalue estimate for the Kohn Laplacian
• Quaternionic contact manifolds with a closed fundamental 4-form
• The Lichnerowicz--Obata theorem for the Kohn Laplacian in three dimensions