Prescribing the Qˉ′\bar Q^{\prime}-Curvature on Pseudo-Einstein CR 3-Manifolds

In this paper we study the problem of prescribing the $\bar Q^{\prime}$-curvature on pseudo-Einstein CR 3-manifolds. In the first stage we study the problem in the compact setting and we show that under natural assumptions, one can prescribe any positive CR pluriharmonic function. In the second stage we study the problem in the non-compact setting of the Heisenberg group. Under mild assumptions on the prescribed function, we prove the existence of a one parameter family of solutions. In fact, we show that one can find two kinds of solutions: normal ones that satisfy an isoperimetric inequality and non-normal ones that have a biharmonic leading term.Comment: 26 page

Some properties of Dirac-Einstein bubbles

We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac-Einstein equations on $\mathbb{R}^3$, which appear in the bubbling analysis of conformal Dirac-Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin-Talenti functions, while the spinorial part is the conformal image of $-\frac{1}{2}$-Killing spinors on the round sphere $\mathbb{S}^3$.Comment: 14 pages. J. Geom. Anal. (2020