## Journal of Differential Geometry

### An Obata-type theorem in CR geometry

#### Abstract

We discuss a sharp lower bound for the first positive eigenvalue of the sublaplacian on a closed, strictly pseudoconvex pseudohermitian manifold of dimension $2m + 1 \geq 5$. We prove that the equality holds iff the manifold is equivalent to the CR sphere up to a scaling. For this purpose, we establish an Obata-type theorem in CR geometry that characterizes the CR sphere in terms of a nonzero function satisfying a certain overdetermined system. Similar results are proved in dimension 3 under an additional condition.

#### Article information

Source
J. Differential Geom., Volume 95, Number 3 (2013), 483-502.

Dates
First available in Project Euclid: 16 October 2013

https://projecteuclid.org/euclid.jdg/1381931736

Digital Object Identifier
doi:10.4310/jdg/1381931736

Mathematical Reviews number (MathSciNet)
MR3128992

Zentralblatt MATH identifier
1277.32038

#### Citation

Li, Song-Ying; Wang, Xiaodong. An Obata-type theorem in CR geometry. J. Differential Geom. 95 (2013), no. 3, 483--502. doi:10.4310/jdg/1381931736. https://projecteuclid.org/euclid.jdg/1381931736