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Wikipedia / CR manifold

In mathematics, a CR manifold, or Cauchy–Riemann 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.

Formally, a CR manifold is a differentiable manifold M together with a preferred complex distribution L, or in other words a complex subbundle of the complexified tangent bundle such that

The subbundle L is called a CR structure on the manifold M-operator; it is defined by

.

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 (or certain real submanifolds of higher codimension) in complex space by studying the properties of holomorphic vector fields which are tangent to the hypersurface.

Suppose for instance that M given by the equation

where z and w. The holomorphic tangent bundle 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 since ∂/∂z and ∂/∂w are linearly independent of their complex conjugates.

More generally, suppose that M with defining equation F(z1, ..., zn) = 0. Then the CR structure L:

for the same reason as before. Moreover, [L,L] ⊂ L since the commutator of holomorphic vector fields annihilating F is again a holomorphic vector field annihilating F.

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 complex 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.

Embedded CR manifolds

Preliminaries

by:

Also relevant are the characteristic annihilators from the Dolbeault complex:

  • In coordinates,

    In coordinates,

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.

Real submanifolds of complex space

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 We say that M is a generic embedded CR submanifold of CR codimensionk. The adjective generic indicates that the tangent space over complex numbers. In most applications, k=1, in which case the manifold is said to be of hypersurface type.

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 nk.

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 (from the side F<0) 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 pseudoconvexity.

This nomenclature comes from the study of pseudoconvex domainss: M if and only if it is (strictly) pseudoconvex as a CR manifold from the side of the domain. (See plurisubharmonic functions and Stein manifold.)

Abstract CR structures

An abstract CR structure on a real manifold M of real dimension n consists of a complex subbundle L of the complexified tangent bundle which is formally integrable, in the sense that [L,L] ⊂ L, which has zero intersection with its complex conjugate. The CR codimension where dimL is the complex dimension. In case k=1, the CR structure is said to be of hypersurface type. Most examples of abstract CR structures are of hypersurface type.

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 by the same expression. The extended form is also sometimes referred to as the Levi form.

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 real; the complex hyperplane a contact form. The conformal class of the contact form is well defined.

. This vector field, called the Reeb vector field, is defined by

.

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.

, then we define the hermitian inner product by setting

.

can be expressed as

.

-direction.

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 John 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. A strictly pseudoconvex CR-manifold also has a canonical Cartan connection, which is defined independently of the choice of contact form. This connection was introduced in 3-dimensions by Cartan, and in higher dimensions by S.S. Chern and J. Moser. The Cartan-Chern-Moser connection and its curvature are invariant under CR-diffeomorphisms.

-operator. It is easy to show that this null--space is an algebra with point-wise multiplication. One can consider both the questions of local embeddability and of global embeddability.

Louis Boutet de Monvel, then the CR-manifold is globally embeddable.

define abstract CR structures on the 3-sphere, which are not globally embeddable. These are often called the Rossi example, though they go back to earlier work of Hans Grauert and appear in a paper by Aldo Andreotti and Yum-Tong Siu. These examples are locally embeddable in a neighborhood of every point, but not globally embeddable: the algebra of CR-functions does not separate points. As shown by Burns are even functions.

Joseph J. Kohn. is CR-invariant.

Note that the set of embeddable perturbations of the CR-structure on a compact 3-dimensional manifold is of infinite dimension and also of infinite co-dimension. Under a positivity condition on the Fourier coefficients of the perturbation term, Daniel Burns and Charles Epstein proved global embeddability for small perturbations of the standard CR structure on the 3-dimensional sphere. can be embedded as a small perturbation of the original embedding. The paper also conjectures a normal form for such perturbations, that includes the positive, embeddable perturbations along with an infinite number of obstructions to embeddability. The existence of this normal form was proved by John Bland.

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 André 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 John 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.

have non-negative Paneitz operators. This is a sort of converse question to the embedding theorems discussed above. In this direction Jeffrey Case, Sagun Chanillo and Paul C. Yang and the CR Yamabe constant of the family of manifolds is uniformly bounded below by a positive constant, then the CR Paneitz operator remains non-negative for the entire family, provided one member of the family has its CR Paneitz operator non-negative. The converse question was finally solved by Yuya Takeuchi. He proved that for embedded, compact CR-3 manifolds that are strictly pseudoconvex, the CR Paneitz operator associated to this embedded manifold is non-negative.

will bound a Complex variety which in general may have singularities. This is the content of the Complex Plateau problem studied in the article by F. Reese Harvey and H. Blaine Lawson. There is also further work on the Complex Plateau problem by Stephen S.-T. Yau.

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 on the Heisenberg group given by

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,

given by

as displayed below:

The work of L. Nirenberg has been extended to a generic result by Howard Jacobowitz and François Trèves. 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 A simplified presentation of Kuranishi's proof is due to Webster.

The problem of local embedding remains open in real dimension 5.

Characteristic ideals

The tangential Cauchy–Riemann complex (Kohn Laplacian, Kohn–Rossi complex)

from the interior to the boundary. The subscript b is to remind one that we are on the boundary. The co-boundary operator takes (0,p) forms to (0,p+1) forms. One can even define the co-boundary operator for an abstract CR manifold even if it is not the boundary of a complex variety. This can be done using the Webster connection.. This complex is called the Tangential Cauchy–Riemann complex or the Kohn–Rossi complex. Investigation of this complex and the study of the Cohomology groups of this complex was done in a fundamental paper by Joseph J. Kohn and Hugo Rossi.

Associated to the Tangential CR complex is a fundamental object in CR Geometry and Several Complex Variables, the Kohn Laplacian. It is defined as:

where the volume form may be derived from a contact form which is associated to the CR structure. See for example the paper of J.M. Lee in the American J. referred below. Note the Kohn Laplacian takes (0,p) forms to (0,p) forms. Functions that are annihilated by the Kohn Laplacian are called CR functions. They are the boundary analogs of holomorphic functions. The real parts of the CR functions are called the CR pluriharmonic functions is a non-negative, formally self-adjoint operator. It is degenerate and has a characteristic set where its symbol vanishes. On a compact, strongly pseudo-convex abstract CR manifold, it has discrete positive eigenvalues which go to infinity and also approach zero. The kernel consists of the CR functions and so is infinite dimensional. If the positive eigenvalues of the Kohn Laplacian are bounded below by a positive constant, then the Kohn Laplacian has closed range and conversely. 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 Kohn Laplacian always has the eigenvalue zero corresponding to the CR functions.

.

and consider the antiholomorphic vector fields which are also group left invariant,

vanishes on functions, we also have the following formula for the Kohn Laplacian for functions on the Heisenberg group:

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:

is the traditional symbol for the sub-Laplacian. Thus

Examples

described above is given by

by

. The Sphere is an example of a CR manifold with constant positive Webster curvature and having zero Webster torsion. The Heisenberg group is an example of a non-compact CR manifold with zero Webster torsion and zero Webster curvature. 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. These spaces can be used as comparison spaces in studying geodesics and volume comparison theorems on CR manifolds with zero Webster torsion akin to the H.E. Rauch comparison theorem in Riemannian Geometry.

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.

See also

Notes

References