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

TL;DR

This paper characterizes the smallest eigenvalue of the sub-Laplacian on a seven-dimensional quaternionic contact manifold, showing it occurs only when the structure is qc-Einstein and qc-equivalent to the 3-Sasakian sphere.

Contribution

It establishes Obata-type eigenvalue theorems for seven-dimensional quaternionic contact manifolds under specific curvature and eigenfunction conditions.

Findings

The lowest eigenvalue is achieved only by qc-Einstein structures.

Such manifolds are qc-equivalent to the standard 3-Sasakian sphere.

Non-negativity of the P-function characterizes the eigenvalue achievement.

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.