The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold in dimension seven

TL;DR

This paper establishes a sharp lower bound for the first eigenvalue of the sub-Laplacian on seven-dimensional quaternionic contact manifolds, extending Lichnerowicz's theorem and characterizing equality cases for 3-Sasakian spheres.

Contribution

It provides the first sharp eigenvalue lower bound for the sub-Laplacian in this setting and characterizes the equality case as the round 3-Sasakian sphere.

Findings

Lower bound for the first eigenvalue under curvature conditions

Equality case characterized by the round 3-Sasakian sphere

Extension of Lichnerowicz's theorem to quaternionic contact manifolds

Abstract

A version of Lichnerowicz' theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the Sp(1)Sp(1) component of the qc-Ricci curvature on a compact seven dimensional quaternionic contact manifold is established. It is shown that in the case of a seven dimensional compact 3-Sasakian manifold the lower bound is reached if and only if the quaternionic contact manifold is a round 3-Sasakian sphere.