CHAPTER 1 Introduction The Riemannian [LP] and CR Yamabe problems [JL1, JL2, JL3, JL4] have been a fruitful subject in geometry and analysis. Major steps in the solutions is the understanding of the conformally flat cases. A model for this setting is given by the corresponding spheres, or equivalently, the Heisenberg groups with, respectively, 0- dimensional and 1-dimensional centers. The equivalence is established through the Cayley transform [K], [CDKR1] and [CDKR2], which in the Riemannian case is the usual stereographic projection. In the present paper we consider the Yamabe problem on the quaternionic Heisenberg group (three dimensional center). This problem turns out to be equiv- alent to the quaternionic contact Yamabe problem on the unit (4n+3)-dimensional sphere in the quaternionic space due to the quaternionic Cayley transform, which is a conformal quaternionic contact transformation (see the proof of Theorem 1.2). The central notion is the quaternionic contact structure (QC structure for short), [Biq1,Biq2], which appears naturally as the conformal boundary at infinity of the quaternionic hyperbolic space, see also [P,GL,FG]. Namely, a QC structure (η, Q) on a (4n+3)-dimensional smooth manifold M is a codimension 3 distribution H, such that, at each point p M the nilpotent Lie algebra Hp (TpM/Hp) is isomorphic to the quaternionic Heisenberg algebra Hm Im H. This is equivalent to the existence of a 1-form η = (η1,η2,η3) with values in R3, such that, H = Ker η and the three 2-forms dηi|H are the fundamental 2-forms of a quaternionic structure Q on H. A special phenomena here, noted by Biquard [Biq1], is that the 3-contact form η determines the quaternionic structure as well as the metric on the horizontal bundle in a unique way. Of crucial importance is the existence of a distinguished linear connection, see [Biq1], preserving the QC structure and its Ricci tensor and scalar curvature Scal, defined in (3.34), and called correspondingly qc-Ricci tensor and qc-scalar curvature. The Biquard connection will play a role similar to the Tanaka-Webster connection [We] and [T] in the CR case. The quaternionic contact Yamabe problem, in the considered setting, is about the possibility of finding in the conformal class of a given QC structure one with constant qc-scalar curvature. The question reduces to the solvability of the Yamabe equation (5.8). As usual if we take the conformal factor in a suitable form the gradient terms in (5.8) can be removed and one obtains the more familiar form of the Yamabe equation. In fact, taking the conformal factor of the form ¯ = u1/(n+1)η reduces (5.8) to the equation Lu 4 n + 2 n + 1 u u Scal = −u2∗−1 Scal, where is the horizontal sub-Laplacian, cf. (5.2), and Scal and Scal are the qc- scalar curvatures correspondingly of (M, η) and (M, ¯), 2∗ = 2Q Q−2 , and Q = 4n +6 1
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