The Sharp Lower Bound of the First Eigenvalue of the Sub-Laplacian on a Quaternionic Contact Manifold |
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Authors: | S Ivanov A Petkov D Vassilev |
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Institution: | 1. Faculty of Mathematics and Informatics, University of Sofia, blvd. James Bourchier 5, 1164, Sofia, Bulgaria 2. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, 87131-0001, USA
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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’s 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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