Vanishing theorem of Kohn–Rossi cohomology class and rigidity of Sasakian space form

Der-Chen Chang (Department of Mathematics and Statistics, Georgetown University, Washington, D.C., U.S.A.; and Graduate Institute of Business Administration, College of Management, Fu Jen Catholic University, New Taipei City, Taiwan)

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

Ting-Jung Kuo (Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan)

Chien Lin (Mathematical Science Research Center, Chongqing University of Technology, Chongqing, China)

Abstract

In this note, under the positivity assumption of the pseudohermitian curvature, we derive the existence theorem for pseudo-Einstein contact forms and rigidity theorems for Sasakian space forms in a closed spherical strictly pseudoconvex CR $3$-manifold of the nonnegative CR Paneitz operator with a kernel consisting of the CR pluriharmonic functions and the CR Q-curvature is pluriharmonic.

Keywords

pseudo-Einstein, CR pluriharmonic operator, CR Paneitz operator, CR Q-curvature, tangential Cauchy–Riemann equation, Sasakian manifold, CR rigidity theorem, Sasakian space form

2010 Mathematics Subject Classification

Primary 32V05, 32V20. Secondary 53C56.

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Der-Chen Chang is partially supported by an NSF grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University.

Shu-Cheng Chang and Ting-Jung Kuo are partially supported in part by the MOST of Taiwan.

Received 7 November 2020

Accepted 19 January 2021

Published 13 May 2022