In mathematics, a CR manifold is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge.
Contents
- Introduction and motivation
- Embedded and abstract CR manifolds
- Preliminaries
- Real submanifolds of complex space
- The Levi form
- Abstract CR structures and Embedding Abstract CR structures in C n displaystyle mathbb C n
- The Levi form and pseudoconvexity
- The tangential CauchyRiemann complex Kohn Laplacian KohnRossi complex
- Examples
- References
Formally, a CR manifold is a differentiable manifold M together with a preferred complex distribution L, or in other words a subbundle of the complexified tangent bundle CTM = TM ⊗ C such that
The bundle L is called a CR structure on the manifold M.
The abbreviation CR stands for Cauchy–Riemann or Complex-Real.
Introduction and motivation
The notion of a CR structure attempts to describe intrinsically the property of being a hypersurface in complex space by studying the properties of holomorphic vector fields which are tangent to the hypersurface.
Suppose for instance that M is the hypersurface of C2 given by the equation
where z and w are the usual complex coordinates on C2. The holomorphic tangent bundle of C2 consists of all linear combinations of the vectors
The distribution L on M consists of all combinations of these vectors which are tangent to M. The tangent vectors must annihilate the defining equation for M, so L consists of complex scalar multiples of
In particular, L consists of the holomorphic vector fields which annihilate F. Note that L gives a CR structure on M, for [L,L] = 0 (since L is one-dimensional) and
More generally, suppose that M is a real hypersurface in Cn, with defining equation F(z1, ..., zn) = 0. Then the CR structure L consists of those linear combinations of the basic holomorphic vectors on Cn:
which annihilate the defining function. In this case,
Embedded and abstract CR manifolds
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 Lagrangian distribution L). Many of the formal geometrical features are similar. 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.
Preliminaries
Embedded CR manifolds are, first and foremost, submanifolds of Cn. Define a pair of subbundles of the complexified tangent bundle C ⊗ TC'n 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
Real submanifolds of complex space
Let M ⊂ Cn be a real submanifold, defined locally as the locus of a system of smooth real-valued functions
F1 = 0, F2 = 0, ..., Fk = 0.Suppose that 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 2n − k. We say that M is an embedded CR manifold of CR codimension k. In most applications, k = 1, in which case the manifold is said to be of hypersurface type.
Let L ⊂ T(1,0)Cn|M be the subbundle of vectors annihilating all of the defining functions F1, ..., Fk. Note that, by the usual considerations for integrable distributions on hypersurfaces, L is involutive. 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.
The Levi form
Let M be a CR manifold of hypersurface type with single defining function F = 0. The Levi form of M, named after Eugenio Elia Levi, is the Hermitian 2-form
This determines a metric on L. M is said to be strictly pseudoconvex if h is positive definite (or pseudoconvex in case h is positive semidefinite). Many of the analytic existence and uniqueness results in the theory of CR manifolds depend on the strict pseudoconvexity of the Levi form.
This nomenclature comes from the study of pseudoconvex domains: M is the boundary of a (strictly) pseudoconvex domain in Cn if and only if it is (strictly) pseudoconvex as a CR manifold. (See plurisubharmonic functions and Stein manifold.)
Abstract CR structures and Embedding Abstract CR structures in C n {displaystyle mathbb {C} ^{n}}
An abstract CR structure on a manifold M of dimension n consists of a subbundle L of the complexified tangent bundle which is formally integrable, in the sense that [L,L] ⊂ L, which is linearly independent of its complex conjugate. The CR codimension of the CR structure is k = n − 2 dim L. In case k = 1, the CR structure is said to be of hypersurface type. Most examples of abstract CR structures are of hypersurface type, unless otherwise made explicit.
The Levi form and pseudoconvexity
Suppose that M is a CR manifold of hypersurface type. The Levi form is the vector valued 2-form, defined on L, with values in the line bundle
given by
h defines a sesquilinear form on L since it does not depend on how v and w are extended to sections of L, by the integrability condition. This form extends to a hermitian form on the bundle
The Levi form can alternatively be characterized in terms of duality. Consider the line subbundle of the complex cotangent bundle annihilating V
For each local section α ∈ Γ(H0M), let
The form hα is a complex-valued hermitian form associated to α.
Generalizations of the Levi form exist when the manifold is not of hypersurface type, in which case the form no longer assumes values in a line bundle, but rather in a vector bundle. One may then speak, not of a Levi form, but of a collection of Levi forms for the structure.
On abstract CR manifolds, of strongly pseudo-convex type, the Levi form gives rise to a pseudo-Hermitian metric. The metric is only defined for the holomorphic tangent vectors and so is degenerate. One can then define a connection and torsion and related curvature tensors for example the Ricci curvature and scalar curvature using this metric. This gives rise to an analogous CR Yamabe problem first studied by David Jerison and Jack Lee.The connection associated to CR manifolds was first defined and studied by Sidney M. Webster in his thesis on the study of the equivalence problem and independently also defined and studied by Tanaka. Accounts of these notions may be found in the articles.
One of the basic questions of CR Geometry is to ask when a smooth manifold endowed with an abstract CR structure can be realized as an embedded manifold in some
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.
In dimension 3, there are obstructions to global embeddability. By taking small perturbations of the standard CR structure on the three sphere
A result of Joseph J. Kohn states that global embeddability is equivalent to the condition that the Kohn Laplacian have closed range. This condition of closed range is not a CR invariant condition.
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 that guarantees global embeddability for abstract strongly pseudo-convex CR structures defined on compact manifolds. Under the hypothesis that the CR Paneitz Operator is non-negative and the CR Yamabe constant is positive, one has global embedding. The second condition can be weakened to a non-CR invariant condition by demanding the Webster curvature of the abstract manifold be bounded below by a positive constant. It allows the authors to get a sharp lower bound on the first positive eigenvalue of Kohn's Laplacian. The lower bound is the analog in CR Geometry of the Andre Lichnerowicz bound for the first positive eigenvalue of the Laplace-Beltrami operator for compact manifolds in Riemannian geometry. Non-negativity of the CR Paneitz operator in dimension 3 is a CR invariant condition as follows by the conformal covariant properties of the CR Paneitz operator on CR manifolds of real dimension 3, first observed by Kengo Hirachi. The CR version of the Paneitz operator, the so-called CR Paneitz Operator first appears in a work of C. Robin Graham and Jack Lee. The operator is not known to be conformally covariant in real dimension 5 and higher, but only in real dimension 3. It is always a non-negative operator in real dimension 5 and higher.
One can ask if all compactly embedded CR manifolds in
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. These results hypothesize assumptions on the Fourier coefficients of the perturbation term.
The realization of the abstract CR manifold as a smooth manifold in some
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). The example of L. Nirenberg may be viewed as a smooth perturbation of the non-solvable complex vector field of Hans Lewy. One can start with the anti-holomorphic vector field
The vector field defined above has two linearly independent first integrals. 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
The CR structure then is seen to be nothing but the restriction of the Complex structure of
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 problem of local embedding remains open in real dimension 5.
The tangential Cauchy–Riemann complex (Kohn Laplacian, Kohn–Rossi complex)
First of all one needs to define a co-boundary operator
Associated to the Tangential CR complex is a fundamental object in CR Geometry and Several Complex Variables, the Kohn Laplacian. It is defined as:
Here
Estimates for
A concrete example of the
Then for a function u we have the (0,1) form
Since
where
are the group left invariant, holomorphic vector fields on the Heisenberg group. The expression for the Kohn Laplacian above can be re-written as follows. First it is easily checked that
Thus we have by an elementary calculation:
The first operator on the right is a real operator and in fact it is the real part of the Kohn Laplacian. It is called the sub-Laplacian. It is a primary example of what is called a Hörmander sums of squares operator. It is obviously non-negative as can be seen via an integration by parts. Some authors define the sub-Laplacian with an opposite sign. In our case we have specifically:
where the symbol
Examples
The canonical example of a compact CR manifold is the real
where
and the almost complex structure on
In recent years, other aspects of analysis on the Heisenberg group have been also studied, like minimal surfaces in the Heisenberg group, the Bernstein problem in the Heisenberg group and curvature flows.