CR manifold

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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.

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 = TMC 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[edit]

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

F(z,w) := |z|^2+|w|^2=1,

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

\frac{\partial}{\partial z},\quad \frac{\partial}{\partial w}.

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

\bar{w}\frac{\partial}{\partial z}-\bar{z}\frac{\partial}{\partial w}.

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 L\cap\bar{L}=\{0\} since ∂/∂z and ∂/∂w are linearly independent of their complex conjugates.

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:

\frac{\partial}{\partial z_1}, ..., \frac{\partial}{\partial z_n}

which annihilate the defining function. In this case, L\cap\bar{L}=\{0\} 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[edit]

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.

Embedded CR manifolds[edit]

Preliminaries[edit]

Embedded CR manifolds are, first and foremost, submanifolds of Cn. Define a pair of subbundles of the complexified tangent bundle C ⊗ TC'n by:

T^{(1,0)}\mathbb{C}^n = \mathrm{span}\left(\frac{\partial}{\partial z_1},\dots,\frac{\partial}{\partial z_n}\right).
T^{(0,1)}\mathbb{C}^n = \mathrm{span}\left(\frac{\partial}{\partial \bar{z}_1},\dots,\frac{\partial}{\partial \bar{z}_n}\right).

Also relevant are the characteristic annihilators from the Dolbeault complex:

  • Ω(1,0)Cn = (T(0,1)Cn). In coordinates,
\Omega^{(1,0)}\mathbb{C}^n = \mathrm{span}(dz_1,\dots,dz_n).
  • Ω(0,1)Cn = (T(1,0)Cn). In coordinates,
\Omega^{(0,1)}\mathbb{C}^n = \mathrm{span}(d\bar{z}_1,\dots,d\bar{z}_n).

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

\partial : \Omega^{(p,q)} \rightarrow \Omega^{(p+1,q)}
\bar{\partial} : \Omega^{(p,q)} \rightarrow \Omega^{(p,q+1)}

Furthermore, there is a decomposition of the usual exterior derivative via d = \partial + \bar{\partial}.

Real submanifolds of complex space[edit]

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:

\partial F_1\wedge\bar{\partial} F_1\wedge\dots \wedge \partial F_k\wedge \bar{\partial} F_k \not= 0.

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[edit]

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,[1] is the Hermitian 2-form

h=i\partial\bar{\partial}F|_L.

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[edit]

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[edit]

Suppose that M is a CR manifold of hypersurface type. The Levi form is the vector valued form, defined on L, with values in the line bundle

V = \frac{TM\otimes{\mathbb C}}{L\oplus\bar{L}}

given by

h(v,w) = \frac{1}{2i}[v,\bar{w}] \mod L\oplus\bar{L},\quad v,w\in L.

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 L\oplus\bar{L} 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

H_0M = V^* = (L\oplus\bar{L})^\perp\sub T^*M\otimes{\mathbb C}.

For each local section α∈Γ(H0M), let

h_\alpha(v,w) = d\alpha(v,\bar{w}) = -\alpha([v,\bar{w}]),\quad v,w\in L\oplus\bar{L}.

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.[2] Accounts of these notions may be found in the articles.[3][4]

One of the basic questions of CR Geometry is to to ask when a smooth manifold endowed with an abstract CR structure can be realized as an embedded manifold in some \mathbb{C}^{n}. Thus not only are we embedding the manifold, but we also demand for global embedding that the map embedding the abstract manifold in \mathbb{C}^n must pull back the induced CR structure of the embedded manifold( coming from the fact that it sits in \mathbb{C}^n) so that the pull back CR structure exactly agrees with the abstract CR structure. Thus global embedding is a two part condition. Here the question splits into two. One can ask for local embeddability or global embeddability.

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.[5]

In dimension 3, there are obstructions to global embeddability. By taking small perturbations of the standard CR structure on the three sphere  \mathbb{S}^3, the resulting abstract CR structure one gets, fails to embed globally. This is sometimes called the Rossi example.[6] The example in fact goes back to Hans Grauert and also appears in a paper by Aldo Andreotti and Yum-Tong Siu.[7]

A result of Joseph J. Kohn states that global embeddability is equivalent to the condition that the Kohn Laplacian have closed range.[8]

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[9] that guarantees global embeddability for abstract strongly pseudo-convex CR structures defined on compact manifolds. Under the hypothesis that the CR Paneitz operator be non-negative and the CR Yamabe constant be 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.[10] 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 first observed by Kengo Hirachi.[11] The CR version of the Paneitz operator first appears in a work of C. Robin Graham and Jack Lee. The operator is not 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.[12]

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.[13]

The realization of the abstract CR manifold as a smooth manifold in some \mathbb{C}^n 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.[14] There is also further work on the Complex Plateau problem by Stephen S.-T. Yau.[15]

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).[16] 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 \overline{L} on the Heisenberg group given by

\overline{L}=\frac{\partial}{\partial \bar{z}}-\imath z\frac{\partial}{\partial t},(z,t)\in\mathbb{C}\times\mathbb{R},\imath=\sqrt{-1}.

The vector field defined above has two linearly independent first integrals. That is there are two solutions to the homogeneous equation,

 \overline{L}Z_i=0,i=1,2, Z_1=z, Z_2=t+\imath |z|^2, dZ_1\wedge dZ_2\not=0.

Since we are in real dimension three the formal integrability condition is simply,

 [\overline{L},\overline{L}]=0

which is automatic. Notice the Levi form is strictly positive definite as a simple calculation gives,

 [\overline{L},L]= 2i\frac{\partial}{\partial t},

where the holomorphic vector field L is given by,

 L=\frac{\partial}{\partial z}+\imath\overline{z}\frac{\partial}{\partial t}.

The first integrals which are linearly independent allow us to realize the CR structure as a graph in \mathbb{C}^2 given by

 (z,t)\to (z,t+\imath |z|^2)

The CR structure then is seen to be nothing but the restriction of the Complex structure of \mathbb{C}^2 to the graph. Nirenberg constructs a single, non-vanishing complex vector field P, defined in a neighborhood of the origin in \mathbb{C}\times\mathbb{R}. He then shows that if Pu=0, then u has to be a constant. Thus the vector field P has no first integrals. The vector field P is created from the anti-holomorphic vector field for the Heisenberg group displayed above by perturbing it by a smooth complex-valued function \phi as displayed below:

 P=\overline{L}+\phi(z,\overline{z},t)\frac{\partial}{\partial t}

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 anyway as a graph in any \mathbb{C}^n. The work of L. Nirenberg has been extended to a generic result by Howard Jacobowitz and Francois Treves.[17] In real dimension 9 and higher, local embedding of abstract CR structures is true by the work of Masatake Kuranishi and in real dimension 7 by the work of Akahori[18] A simplified presentation of Kuranishi's proof is due to Webster.[19]

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

Characteristic ideals[edit]

The tangential Cauchy–Riemann complex(Kohn Laplacian, Kohn-Rossi Complex)[edit]

First of all one needs to define a co-boundary operator \overline{\partial_b}. For CR manifolds that arise as boundaries of complex manifolds, one may view this operator as the restriction of \overline{\partial} 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.[20] The co-boundary operator \overline{\partial_b} forms a complex, that is \overline{\partial_b}\circ\overline{\partial_b}=0. 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. [21]

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

 \Box_b=\overline{\partial_b}\overline{\partial_b}^\star+\overline{\partial_b}^\star\overline{\partial_b}

Here \overline{\partial_b}^\star denotes the formal adjoint of \overline{\partial_b} with respect to L^2(M) 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. The Kohn Laplacian  \Box_b 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.

Estimates for \Box_b and \overline{\partial_b} have been obtained in various function spaces in various settings. These estimates are easiest to derive when the manifold is strongly pseudoconvex, for then one can replace the manifold by osculating it to a high enough order with the Heisenberg group. Then using the group property and attendant convolution structure of the Heisenberg group, one can write down inverses/parametrices or relative parametrices to  \Box_b.[22]

A concrete example of the \overline{\partial_b} operator can be provided on the Heisenberg group. Consider the general Heisenberg group \mathbb{C}^n\times \mathbb{R} and consider the antiholomorphic vector fields which are also group left invariant,

 \overline{L}_j= \frac{\partial}{\partial \overline{z_j}}-\imath z_j\frac{\partial}{\partial t}, j=1,2,\cdots ,n, (z_1,z_2,\cdots, z_n)\in\mathbb{C}^n, t\in \mathbb{R}.

Then for a function u we have the (0,1) form \omega

 \omega =\overline{\partial_b}u=\sum_{j=1}^n \overline{L_j}u\ d\overline{z_j}.

Since \overline{\partial_b}^\star vanishes on functions, we also have the following formula for the Kohn Laplacian for functions on the Heisenberg group:

 \Box_b=-\sum_{j=1}^n L_j \overline{L_j}

where

L_j=\frac{\partial}{\partial z_j}+\imath\overline{z_j}\frac{\partial}{\partial t},

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

 [L_j,\overline{L_j}]=-2\imath T, T=\frac{\partial}{\partial t},j=1,2,\cdots,n

Thus we have by an elementary calculation:

 \Box_b=-\frac{1}{2}\sum_{j=1}^n(L_j\overline{L_j}+\overline{L_j}L_j)+\imath nT

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 Hormander sums of squares operator.[23][24] 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:

\Delta_b=-\frac{1}{2}\sum_{j=1}^n (L_j\overline{L_j}+\overline{L_j}L_j)

where the symbol \Delta_b is the traditional symbol for the sub-Laplacian. Thus

\Box_b=\Delta_b+\imath nT

Examples[edit]

The canonical example of a CR manifold is the real 2n+1 sphere as a submanifold of \mathbb{C}^{n+1}. The bundle L described above is given by

L = \mathbb{C}TS^{2n+1} \cap T^{1,0}\mathbb{C}^{n+1}

where T^{1,0}\mathbb{C}^{n+1} is the bundle of holomorphic vectors. The real form of this is given by P=\Re (L\oplus \bar{L}), the bundle given at a point p\in S^{2n+1} concretely in terms of the complex structure, I, on \mathbb{C}^{n+1} by

P_p = \{ X\in T_pS^{2n+1} : IX \in T_pS^{2n+1}\subset T_p\mathbb{C}^{n+1}\},

and the almost complex structure on P is just the restriction of I. 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 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.[25]

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.[26]

See also[edit]

Notes[edit]

  1. ^ See (Levi 909, p. 207): the Levi form is the differential form associated to the differential operator C, according to Levi's notation.
  2. ^ Tanaka, N. (1975). "A Differential Geometric Study on Strongly Pseudoconvex Manifolds". Lectures in Mathematics, Kyoto University (Tokyo: Kinokuniya Book Store) 9. 
  3. ^ Lee, John,M. (1988). "Pseudo-Einstein Structures on CR manifolds". American Journal of Mathematics 110 (1): 157–178. doi:10.2307/2374543. 
  4. ^ Webster, Sidney, M. (1978). "Pseudo-hermitian Structures on a Real Hypersurface". Journal of Differential Geometry 13: 25–41. 
  5. ^ Boutet de Monvel, Louis (1974). "Integration de equations Cauchy-Riemann induites formelle". Seminaire Equations aux derivees Partielles (Ecole Polytechnique) 9: 1–13. 
  6. ^ Chen, S.-C.; Shaw, Mei-Chi (2001). Partial Differential Equations in Several Complex Variables. 19, AMS/IP Studies in Advanced Mathematics. Providence, RI: AMS. 
  7. ^ Andreotti, Aldo and Siu, Yum-Tong (1970). "Projective Embedding of Pseudoconcave Spaces". Annali della Scuola Norm. Sup. Pisa, Classe di Scienze 24 (5): 231–278. 
  8. ^ Kohn, Joseph, J. (1986). "The Range of the Tangential Cauchy-Riemann Operator". Duke Mathematical Journal 53: 525–545. doi:10.1215/S0012-7094-86-05330-5. 
  9. ^ Chanillo, Sagun, Chiu, Hung-Lin and Yang, Paul C. (2012). "Embeddability for 3-dimensional CR manifolds and CR Yamabe Invariants". Duke Mathematical Journal 161 (15): 2909–2921. doi:10.1215/00127094-1902154. 
  10. ^ Lichnerowicz, Andre (1958). Ge'ome'trie des Groupes de transformations. Paris: Dunod. 
  11. ^ Hirachi, Kengo (1993). "Scalar Pseudo-hermitian Invariants and the Szeg\"o kernel on three dimensional CR manifolds". Complex Geometry(Osaka 1990)Lecture Notes in Pure and Applied Math. (New York: Marcel Dekker) 143: 67–76. 
  12. ^ Graham, C. Robin and Lee, John, M. (1988). "Smooth Solutions of Degenerate Laplacians on Strictly Pseudo-convex Domains". Duke Mathematical Journal 57: 697–720. doi:10.1215/S0012-7094-88-05731-6. 
  13. ^ Burns, Daniel, M. and Epstein, Charles, L. (1990). "Embeddability for Three dimensional CR manifolds". J. of the American Math. Soc. 3: 809–841. doi:10.1090/s0894-0347-1990-1071115-4. 
  14. ^ Harvey, F.R. and Lawson, H.B., Jr. (1978). "On boundaries of complex analytic varieties I". Ann. Math. 102 (2): 223–290. doi:10.2307/1971032. JSTOR 1971032. 
  15. ^ Yau, Stephen S.-T. (1981). "Kohn-Rossi Cohomology and its Application to the Complex Plateau Problem I". Annals of Math. 113: 67–110. doi:10.2307/1971134. 
  16. ^ Nirenberg, Louis (1974). "On a Question of Hans Lewy". Russian Math. Surveys 29: 251–262. doi:10.1070/rm1974v029n02abeh003856. 
  17. ^ Jacobowitz, Howard and Treves, Jean-Francois (1982). "Non Realizable CR Structures". Inventiones Math. 66: 231–250. doi:10.1007/bf01389393. 
  18. ^ Akahori, Takao (1987). "A New approach to the Local Embedding theorem of CR Structures of n\geq 4(the local solvability of the operator \overline{\partial_b} in the abstract sense)". Memoirs of the American Math. Society 67 (366). 
  19. ^ Webster, Sidney, M. (1989). "On the Proof of Kuranishi's Embedding Theorem". Annales Institut Henri Poincaré(C) Analyse Nonlineare 6 (3): 183–207. 
  20. ^ Lee, John M. (1986). "The Fefferman metric and pseudo-hermitian invariants". Transactions of the Amer. Math. Soc. 296: 411–429. doi:10.1090/s0002-9947-1986-0837820-2. 
  21. ^ Kohn, Joseph J. and Rossi, Hugo (1965). "On the Extension of Holomorphic functions from the boundary of Complex Manifolds". Annals of Math. 81: 451–472. doi:10.2307/1970624. 
  22. ^ Greiner, P.C. and Stein, E. M. (1977). Estimates for the \overline{\partial}-Neumann problem. Mathematical Notes 19 (Princeton Univ. Press). 
  23. ^ Hormander, Lars (1967). "Hypoelliptic second-order differential equations". Acta Math. 119: 147–171. doi:10.1007/bf02392081. 
  24. ^ Kohn, Joseph J. (1972). "Subelliptic estimates". Proceedings Symp. in Pure Math.(AMS) 35: 143–152. 
  25. ^ Chanillo, Sagun and Yang, Paul C. (2009). "Isoperimetric and Volume Comparison theorems on CR manifolds". Annali della Scuola Norm. Sup. Pisa, Classe di Scienze 8 (2): 279–307. doi:10.2422/2036-2145.2009.2.03. 
  26. ^ Capogna, Luca; Danielli, Donatella; Pauls, Scott; Tyson, Jeremy (2007). "Applications of Heisenberg Geometry". An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem. Progress in Mathematics 259 (Berlin: Birkhauser). pp. 45–48. 

References[edit]

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