The Yamabe problem on quaternionic contact manifolds

Abstract

By constructing normal coordinates on a quaternionic contact manifold M, we can osculate the quaternionic contact structure at each point by the standard quaternionic contact structure on the quaternionic Heisenberg group. By using this property, we can do harmonic analysis on general quaternionic contact manifolds, and solve the quaternionic contact Yamabe problem on M if its Yamabe invariant satisfies λ(M) < λ(H n ).

On the tangential Cauchy‐Fueter operators on nondegenerate quadratic hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {H}^2}$\end{document}

“…(3) There exists a class of hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^n$\end{document} with the quaternionic contact structure 5, which can be approximated by the (right) quaternionic Heisenberg group pointwisely 16.…”

“…It is easy to see that the restriction to ∂Ω of a function regular in a neighborhood of ∂Ω is a Cauchy‐Fueter function, and so Cauchy‐Fueter functions are abundant. Cauchy‐Fueter functions have already been applied to determine the extremals for the Sobolev inequality on the quaternionic Heisenberg group and to solve the quaternionic contact Yamabe problem (see 12, 16 and references therein). Suppose \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\overline{\partial }_{q_n}\rho \ne 0$\end{document} locally.…”

The qc Yamabe problem on non-spherical quaternionic contact manifolds

“…The Biquard connection plays a role similar to the Tanaka-Webster connection [31] and [29] in the CR case. A natural question coming from the conformal freedom of the qc structures is the quaternionic contact Yamabe problem [30,15,20]: The qc Yamabe problem on a compact qc manifold M is the problem of finding a metric g ∈ [g] on H for which the qc-scalar curvature is constant.…”

“…[30,15,20,21]). Given a quaternionic contact (qc) manifold (M, [η]) with a fixed conformal class defined by a quaternionic contact form η, solutions to the quaternionic contact Yamabe problem are critical points of the qc Yamabe functional If η is a fixed qc contact form one considers the functional (which is also called qc Yamabe functional ) The main result of W. Wang [30] states that the qc Yamabe constant of a compact quaternionic contact manifold is always less or equal than that of the standard 3-Sasakian sphere, λ(M ) ≤ λ(S 4n+3 ) and, if the constant is strictly less than that of the sphere, the qc Yamabe problem has a solution, i.e. there exists a global qc conformal transformation sending the given qc structure to a qc structure with constant qc scalar curvature.…”

On seven-dimensional quaternionic contact solvable Lie groups

“…Such structures have been considered in connection with the quaternionic contact Yamabe problem [10,11,14,15]. Results about the CR-structure on the twistor space of a qc manifold were given in [2,4,8,7].…”