A positive mass theorem in three dimensional Cauchy–Riemann geometry

Jih Hsin Cheng, Andrea Malchiodi, Paul Yang

Research output: Contribution to journalArticle

10 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)276-347
Number of pages72
JournalAdvances in Mathematics
Volume308
DOIs
StatePublished - Feb 21 2017

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • CR Paneitz operator
  • CR Yamabe problem
  • CR geometry
  • Positive mass theorem
  • Tanaka–Webster curvature

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