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Prescribing The Q¯ ′ -Curvature On Pseudo-Einstein Cr 3-Manifolds, Ali Maalaoui

Mathematics

In this paper we study the problem of prescribing the Q ¯ ′ -curvature on embeddable 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 (resp. negative) CR pluriharmonic function, if ∫ M Q ′ d v θ > 0 (resp. ∫ M Q ′ d v θ < 0 ). 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 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.

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