[1][2] The notion of a CR structure attempts to describe intrinsically the property of being a hypersurface (or certain real submanifolds of higher codimension) in complex space by studying the properties of holomorphic vector fields which are tangent to the hypersurface.
Then the CR structure L consists of those linear combinations of the basic holomorphic vectors on
Moreover, [L,L] ⊂ L since the commutator of holomorphic vector fields annihilating F is again a holomorphic vector field annihilating F. There is a sharp contrast between the theories of embedded CR manifolds (hypersurface and edges of wedges in complex space) and abstract CR manifolds (those given by the complex distribution L).
These include: Embedded CR manifolds possess some additional structure, though: a Neumann and Dirichlet problem for the Cauchy–Riemann equations.
This article first treats the geometry of embedded CR manifolds, shows how to define these structures intrinsically, and then generalizes these to the abstract setting.
by: Also relevant are the characteristic annihilators from the Dolbeault complex: The exterior products of these are denoted by the self-evident notation Ω(p,q), and the Dolbeault operator and its complex conjugate map between these spaces via: Furthermore, there is a decomposition of the usual exterior derivative via
be a real submanifold, defined locally as the locus of a system of smooth real-valued functions Suppose that the complex-linear part of the differential of this system has maximal rank, in the sense that the differentials satisfy the following independence condition: Note that this condition is strictly stronger than needed to apply the implicit function theorem: in particular, M is a manifold of real dimension
Moreover, the independence condition implies that L is a bundle of constant rank n − k. Henceforth, suppose that k = 1 (so that the CR manifold is of hypersurface type), unless otherwise noted.
[4] Many of the analytic existence and uniqueness results in the theory of CR manifolds depend on the pseudoconvexity.
On abstract CR manifolds, of strongly pseudo-convex type, the Levi form gives rise to a pseudo-Hermitian metric.
This gives rise to an analogous CR Yamabe problem first studied by David Jerison and John Lee.
must pull back the induced CR structure of the embedded manifold( coming from the fact that it sits in
Global embeddability is always true for abstractly defined, compact CR structures which are strongly pseudoconvex, that is the Levi form is positive definite, when the real dimension of the manifold is 5 or higher by a result of Louis Boutet de Monvel.
[9] The example in fact goes back to Hans Grauert and also appears in a paper by Aldo Andreotti and Yum-Tong Siu.
In dimension 3, a non-perturbative set of conditions that are CR invariant has been found by Sagun Chanillo, Hung-Lin Chiu and Paul C. Yang[12] that guarantees global embeddability for abstract strongly pseudo-convex CR structures defined on compact manifolds.
In this direction Jeffrey Case, Sagun Chanillo and Paul C. Yang have proved a stability theorem.
[17] There are also results of global embedding for small perturbations of the standard CR structure for the 3-dimensional sphere due to Daniel Burns and Charles Epstein.
This is the content of the Complex Plateau problem studied in the article by F. Reese Harvey and H. Blaine Lawson.
[20] Local embedding of abstract CR structures is not true in real dimension 3, because of an example of Louis Nirenberg(the book by Chen and Mei-Chi Shaw referred below also carries a presentation of Nirenberg's proof).
[21] The example of L. Nirenberg may be viewed as a smooth perturbation of the non-solvable complex vector field of Hans Lewy.
That is there are two solutions to the homogeneous equation, Since we are in real dimension three the formal integrability condition is simply, which is automatic.
Notice the Levi form is strictly positive definite as a simple calculation gives, where the holomorphic vector field L is given by, The first integrals which are linearly independent allow us to realize the CR structure as a graph in
is created from the anti-holomorphic vector field for the Heisenberg group displayed above by perturbing it by a smooth complex-valued function
as displayed below: Thus this new vector field P, has no first integrals other than constants and so it is not possible to realize this perturbed CR structure in any way as a graph in any
The work of L. Nirenberg has been extended to a generic result by Howard Jacobowitz and François Trèves.
[22] In real dimension 9 and higher, local embedding of abstract strictly pseudo-convex CR structures is true by the work of Masatake Kuranishi and in real dimension 7 by the work of Akahori[23] A simplified presentation of Kuranishi's proof is due to Webster.
One can even define the co-boundary operator for an abstract CR manifold even if it is not the boundary of a complex variety.
On a compact, strongly pseudo-convex abstract CR manifold, it has discrete positive eigenvalues which go to infinity and also approach zero.
Thus for embedded CR structures using the result of Kohn stated above, we conclude that the compact CR structure that is strongly pseudoconvex is embedded if and only if the Kohn Laplacian has positive eigenvalues that are bounded below by a positive constant.
The unit circle bundle over compact Riemann surfaces with genus strictly greater than 1 also provides examples of CR manifolds which are strongly pseudoconvex and have zero Webster torsion and constant negative Webster curvature.