Asian Journal of Mathematics

Volume 22 (2018)

Number 5

On the CR analogue of Reilly formula and Yau eigenvalue conjecture

Pages: 919 – 940

DOI: https://dx.doi.org/10.4310/AJM.2018.v22.n5.a6

Authors

Shu-Cheng Chang (Department of Mathematics and Taida Institute for Mathematical Sciences (TIMS), National Taiwan University, Taipei, Taiwan)

Chih-Wei Chen (Department of Applied Mathematics, National Sun Yet-sen University, 80424, Taiwan)

Chin-Tung Wu (Department of Applied Mathematics, National Pingtung University, Pingtung, Taiwan)

Abstract

In this paper, we derive the CR Reilly’s formula and its applications to studying of the first eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded $p$-minimal hypersurfaces. In particular, we obtain the first Dirichlet eigenvalue estimate in a compact pseudohermitian $(2n+1)$-manifold with boundary and the first eigenvalue estimate of the tangential sublaplacian on closed oriented embedded $p$-minimal hypersurfaces in a closed pseudohermitian $(2n+1)$-manifold of vanishing torsion.

Keywords

pseudohermitian minimal surface, CR Dirichlet eigenvalue, CR Reilly formula, tangential sublaplacian, CR Yau eigenvalue conjecture

2010 Mathematics Subject Classification

53C56, 32V05, 32V20

Received 5 September 2015

Accepted 25 November 2016

Published 9 November 2018