Smooth Quasi--Einstein Lifts of Locally Embeddable Three-Dimensional CR Manifolds

Abstract

We prove a $C^\infty$ analogue of the real-analytic quasi--Einstein lifting theorem of Hill, Lewandowski and Nurowski \cite[Theorem~2.5]{HLN2008}. Every smooth, locally CR-embeddable, strictly pseudoconvex three-dimensional CR manifold admits, for each real cosmological constant, a local smooth Lorentzian quasi--Einstein lift whose null congruence is geodesic, shearfree and twisting and whose Weyl tensor has Petrov type~$\mathrm{II}$ or $\mathrm{D}$. The result extends local lifting beyond the Sasakian setting of the global constructions in \cite{GSS2020} and the local results in \cite{GGS2026}. The proof establishes smooth local solvability of the reduced semilinear contact equation by combining an $L^\infty$ contraction with subelliptic regularity.