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A positive mass theorem in three dimensional Cauchy-Riemann geometry

Author(s): Cheng, Jih-Hsin; Malchiodi, Andrea; Yang, Paul C

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Abstract: We define an ADM-like mass, called p-mass, for an asymptotically flat pseudohermitian manifold. The p-mass for the blow-up of a compact pseudohermitian manifold (with no boundary) is identified with the first nontrivial coefficient in the expansion of the Green function for the CR Laplacian. We deduce an integral formula for the p-mass, and we reduce its positivity to a solution of Kohn’s equation. We prove that the p-mass is non-negative for (blow-ups of) compact 3-manifolds of positive CR Yamabe invariant and with non-negative CR Paneitz operator. Under these assumptions, we also characterize the zero mass case as the standard three dimensional CR sphere. We then show the existence of (non-embeddable) CR 3-manifolds having nonpositive Paneitz operator or negative p-mass through a second variation formula. Finally, we apply our main result to find solutions of the CR Yamabe problem with minimal energy. (C) 2016 Elsevier Inc. All rights reserved.
Publication Date: 21-Feb-2017
Electronic Publication Date: 2-Jan-2017
Citation: Cheng, Jih-Hsin, Malchiodi, Andrea, Yang, Paul. (2017). A positive mass theorem in three dimensional Cauchy-Riemann geometry. ADVANCES IN MATHEMATICS, 308 (276 - 347. doi:10.1016/j.aim.2016.12.012
DOI: doi:10.1016/j.aim.2016.12.012
ISSN: 0001-8708
EISSN: 1090-2082
Pages: 276 - 347
Language: English
Type of Material: Journal Article
Journal/Proceeding Title: ADVANCES IN MATHEMATICS
Version: Author's manuscript

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