On the estimate of the first eigenvalue of a sublaplacian on a pseudohermitian 3-manifold

Shu-Cheng Chang and Hung-Lin Chiu

Vol. 232 (2007), No. 2, 269–282

DOI: 10.2140/pjm.2007.232.269

Abstract

In this paper, we study a lower bound estimate of the first positive eigenvalue of the sublaplacian on a three-dimensional pseudohermitian manifold. S.-Y. Li and H.-S. Luk derived the lower bound estimate under certain conditions for curvature tensors bounded below by a positive constant. By using the Li–Yau gradient estimate, we are able to get an effective lower bound estimate under a general curvature condition. The key is the discovery of a new CR version of the Bochner formula which involves the CR Paneitz operator.

Keywords

eigenvalue, gradient estimate, pseudohermitian manifold, Tanaka–Webster curvature, pseudohermitian torsion, CR Paneitz operator, sublaplacian, Carnot–Carathéodory distance, diameter

Mathematical Subject Classification 2000

Primary: 32V05, 32V20

Secondary: 53C56

Milestones

Received: 25 April 2006

Accepted: 16 January 2007

Published: 1 October 2007