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

TL;DR

This paper establishes a lower bound for the first eigenvalue of the sub-Laplacian on quaternionic contact manifolds using a Bochner formula, with applications to sharp inequalities and characterizations of special geometries.

Contribution

It introduces a Bochner type formula for the sub-Laplacian on quaternionic contact manifolds and derives a Lichnerowicz type eigenvalue estimate under curvature bounds.

Findings

Lower bound for eigenvalues on quaternionic contact manifolds.

Equality case characterizes round 3-Sasakian spheres.

Sharp inequality for horizontal Hessian on quaternionic Heisenberg group.

Abstract

The main technical result of the paper is a Bochner type formula for the sub-laplacian on a quaternionic contact manifold. With the help of this formula we establish a version of Lichnerowicz' theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the $Sp(n)Sp(1)$ components of the qc-Ricci curvature. It is shown that in the case of a 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is a round 3-Sasakian sphere. Another goal of the paper is to establish a-priori estimates for square integrals of horizontal derivatives of smooth compactly supported functions. As an application, we prove a sharp inequality bounding the horizontal Hessian of a function by its sub-Laplacian on the quaternionic Heisenberg group.