Pseudo-Hermitian Geometry in 3-D

Pseudo-hermitian geometry is the study of contact form in conjunction with an almost complex structure on the contact planes. In this two part survey, we first discuss the theory of surfaces in 3-D pseudo-hermitian manifolds. The local geometry of the surface is governed by the p-mean curvature equation. In analogy with the Gauss and Codazzi equation of a surface in Euclidean three space, we develop their analogue of these two equations. Due to the degeneracy of the p-mean curvature equation, the regularity theory is quite interesting. In the second part of this article, we describe a CR-invariant condition for the embeddability question. In CR geometry in 3-D, there is no local integrability condition for the almost-complex structure, so the condition to impose is global in nature. The condition involves two geometrically defined linear operators which have their counterparts in conformal geometry. We also formulate a notion of CR mass and discuss a positive mass theorem to characterize the Heisenberg space. The solvability of the \({\bar\partial}\) Neumann problem is also a crucial ingredient in the positivity of the CR mass.