# A positive mass theorem in three dimensional Cauchy–Riemann geometry

Jih Hsin Cheng, Andrea Malchiodi, Paul Yang

Research output: Contribution to journalArticlepeer-review

15 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 language English (US) 276-347 72 Advances in Mathematics 308 https://doi.org/10.1016/j.aim.2016.12.012 Published - 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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