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

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

2011-12-04

arxiv.org/abs/1112.0779v1

Mathematics

Differential Geometry

15 pages, LaTeX2e

Scientific paper

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.

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