The authors provide an explicit lower bound for all eigenvalues of the sub-Laplacian operator \(\Delta\) defined in a seven-dimensional compact quaternionic contact (abridged by qc) manifold \((M,g,\mathbb Q)\). Loosely speaking, this means that \(M\) is a seven-dimensional manifold endowed with a smooth section \(g\) of positive definite quadratic form on the three-codimensional horizontal distribution \(H\) (equipped with an explicit Lie group structure), and \(\mathbb Q\) is a vector bundle over \(M\) of rank three and made up by endomorphisms of \(H\). Locally, \(\mathbb Q\) is similar to the set of unit quaternions. The case of a compact qc-manifold of dimension bigger than seven was already treated by the same authors. From the Biquards connection, the authors define the qc-Ricci tensor on the section of the tangent bundle \(TM\), also they define the normalized qc-scalar, denoted by \(S\). Accordingly, on \(H\) they provide the qc-Lichnerowicz type inequality. The authors also set up the \(P\)-function of a smooth function defined on \(M\) and it is given in terms of the third-order covariant derivative \(S\), the three generators of \(\mathbb Q\) and the torsion which is defined explicitly.
As an application, they show the positivity of the \(P\)-function of any eigenfunction of \(\Delta\) defined on a seven-dimensional compact qc-Einstein manifold with \(S=2\). Thus, the principal theorem 1.1 states that if the qc-Lichnerowicz type inequality is satisfied and the \(P\)-function of each eigenfunction is positive, then its associated eigenvalue has an accurate lower bound.