An Invitation to Cauchy-Riemann and Sub-Riemannian Geometries John P

An Invitation to Cauchy-Riemann and Sub-Riemannian Geometries John P. D’Angelo and Jeremy T. Tyson

certain amount of complex analysis, in both one and several variables, com- bines with some differential geometry to form the basis of research in diverse parts of mathematics. We first describe Athat foundation and then invite the reader to follow specific geometric paths based upon it. We discuss such topics as the failure of the Riemann mapping theorem in several variables, the geometry of real hypersurfaces in complex Euclidean space, the Heisenberg group, sub-Riemannian manifolds and their metric space structure, the Hopf fibration, sub-Riemannian geodesics on the three-sphere, CR mappings invariant under a finite group, and finite type conditions. All of these topics fall within the branch of mathematics described by the title, but of course they form only a small part of it. We hope that the chosen topics are representative and appealing. The connections among them provide a fertile ground for future research. We modestly hope that this article inspires others to develop these connections further. To this end, our refer- Figure 1. Georg Friedrich Bernhard Riemann. ence list includes a diverse collection of books and accessible articles. Given Riemann’s contributions to geometry, it There Is No Riemann Mapping Theorem in is not surprising that his name occurs twice in the Higher Dimensions title. It is more surprising that the uses of his name In an attempt to understand complex analysis here come from different parts of mathematics. We in several variables, especially questions revolv- find it delightful that contemporary mathematics ing around the failure of the Riemann mapping is forging connections that even Riemann could theorem, many mathematicians have focused at- not have anticipated. tention on geometric properties of the boundaries John P. D’Angelo is professor of mathematics at the Uni- of domains and how those geometric properties versity of Illinois. His email address is [email protected]. influence the complex analysis on the domain. edu. Such investigations have led to the fields of CR Jeremy T. Tyson is associate professor of mathemat- geometry and sub-Riemannian geometry. ics at the University of Illinois. His email address is We begin with the Riemann mapping theo- [email protected]. rem. Let Ω be an open, connected, and simply

208 Notices of the AMS Volume 57, Number 2 connected proper subset of C. Then there is a is no biholomorphic map from any polydisc to bijective holomorphic mapping f from Ω to the unit any ball. One classical proof of this statement disc D. The inverse mapping is also holomorphic. involves showing that the groups of biholomorphic We say that Ω and D are biholomorphically equiva- automorphisms of the ball and of the polydisc lent. The entire plane must of course be excluded, have different dimensions. because a bounded holomorphic function on C re- A second proof (see [D1, page 14]) proceeds in the duces to a constant. This theorem has applications following manner. First consider a biholomorphic throughout both pure and applied mathematics. In (or even proper) image Ω of the Cartesian product of many of these applications, such as uniform fluid bounded domains; one shows that bΩ must contain flow, the upper half plane substitutes for the unit complex analytic sets. Then an easy computation, disc. The explicit linear fractional transformation given in Lemma 6, shows that the definiteness of f (the Cayley transformation) defined by the Levi form precludes the existence of positive- i − w dimensional complex analytic subsets of b . This (1) z = f (w) = Ω i + w computation helps unify several of the ideas in this maps the upper half plane biholomorphically to article. For example, higher order commutators of the unit disc. complex vector fields can determine obstructions We mention in passing one of the most funda- to the existence of complex analytic sets in a real mental geometric generalizations of the Riemann hypersurface. Such obstructions lead to geometric mapping theorem, although we will not head in finite type conditions for subelliptic estimates. In that direction. The uniformization theorem states two complex dimensions a subelliptic estimate (see that the universal cover of a compact Riemann page 217) holds if and only if the bracket-generating surface must be one of three objects: the unit disc property (see page 212) holds. D, the complex plane C, or the Riemann sphere C. Consider a domain Ω with smooth boundary As a reference for analysis in one complex variable, bΩ. In order that geometric results about bΩ be we need only mention the classical text by Ahlfors meaningful for complex analysis on Ω, one requires [Ahl]. results concerning the boundary smoothness of Let us now consider complex Euclidean space Cn biholomorphic maps. Two early references in this of arbitrary dimension. We equip Cn with the usual area are [Bel] and [Fef]; we do not attempt to list any P Hermitian inner product given by hz, wi = zj w j of the many newer references. In general, smoothly and the corresponding norm given by |z| = hz, zi1/2. bounded domains, topologically equivalent to The unitary group U(n) consists of linear mappings the unit ball Bn, are seldom biholomorphically preserving the inner product. The unit ball Bn is equivalent to it. On the other hand, Fridman [Fri] the set of z for which |z| < 1. has established a wonderful approximate Riemann In dimension at least two the unit ball is not mapping theorem: given any topological ball and a biholomorphically equivalent to a half space. There positive , there is a biholomorphic mapping to a are many possible proofs. One approach takes into domain whose boundary is always within distance account the geometry of the boundaries. When the of the unit sphere. boundaries are smooth manifolds (as they are in this case), the Levi form plays a crucial role. The The Unit Sphere and the Heisenberg Group Levi form for a real hypersurface in Cn (for n ≥ 2) The quintessential principle connecting CR geom- is the complex variable analogue of the second etry and sub-Riemannian geometry arises from n fundamental form for a real hypersurface in R . the correspondence between the unit sphere bBn We define and discuss the Levi form on page 211. and the Heisenberg group, which bounds the

For now we simply note that the Levi form is unbounded Siegel upper half space Hn. We will n−1 a Hermitian form defined on part of the tangent obtain a useful identification of bHn with C × R, spaces of the boundary. There is no Levi form in on which we will impose a nonabelian group law one dimension, because the tangent space is one arising from the biholomorphic automorphisms of real dimension and the part on which the Levi form Hn. The equivalence between Hn and Bn generalizes acts does not exist. For a half plane in dimension the one-dimensional Cayley transformation. at least 2 the Levi form is identically zero. For The Siegel upper half space Hn is the set of w the unit sphere it is positive definite. As a result in Cn such that the boundary geometries are so different that no n−1 biholomorphic mapping can exist. By contrast, X 2 (2) Im(wn) > |wj | . in one dimension, different boundary curves are j=1 indistinguishable from this point of view. A polydisc is a Cartesian product of discs. For The following elementary but useful identity is the 2 key to discovering the biholomorphic map: example, the subset of C defined by |z1| < 1 and 2 2 |z2| < 1 is a polydisc. It is also not biholomor- i + ζ i − ζ (3) Im(ζ) = − . phically equivalent to B2. In fact, for n ≥ 2 there 2 2

February 2010 Notices of the AMS 209 With ζ = wn we plug (3) into (2) and rewrite the CR Structures result to obtain We next discuss the geometry of a general real n 2 n−1 2 hypersurface M in C . The beautiful interplay i + wn X i − wn (4) > |w |2 + . between real and complex geometry dominates the 2 j 2 j=1 discussion. See [BER], [D1], [DT], [Jac], [Tre] and their references for more about CR geometry. After dividing by | i+wn |2 and changing notation, 2 We wish to consider complex vector ﬁelds and inequality (4) becomes therefore start by considering the complexiﬁed n X tangent bundle 1 > |z |2. j n = n ⊗ j=1 (6) CT(C ) T(C ) C.

2wj i−wn As usual in complex analysis we define the complex Here zj = for j < n and zn = . The reader i+wn i+wn partial derivative operators should check that this transformation f : w , z is ! ∂ 1 ∂ ∂ biholomorphic from Hn to Bn and also compare it = − i with the transformation in (1). ∂zj 2 ∂xj ∂yj and ! The automorphism group of Hn ∂ 1 ∂ ∂ = + i . We alluded earlier to the biholomorphic automor- ∂zj 2 ∂xj ∂yj phism groups of the ball and the polydisc. For any Using these operators we decompose the exterior complex manifold the automorphism group is a derivative d as fundamental algebraic invariant. For example, the n n X ∂f X ∂f automorphism group of the upper half plane in (7) df = ∂f + ∂f = dz + dz . ∂z j ∂z k C is PSL(2, R). Its elements act on the (one-point j=1 j k=1 k compactification of the) boundary of the upper half 2 Since d2 = 0, it follows that ∂2 = ∂ = ∂∂ + ∂∂ = 0. plane as linear fractional transformations x , ax+b , cx+d A section of the bundle (6) is a complex vector − = for ad bc 1. Any such transformation can be field. Each such vector field is a combination of represented as the composition of dilations x , rx, both z and z derivatives with smooth coefficients: translations x , x + x0, and possibly the inversion n n 1 X ∂ X ∂ x , x . Automorphisms fixing the point at infinity L = aj (z) + bj (z) . do not require the inversion. j=1 ∂zj j=1 ∂zj We now describe the analogous situation on We obtain two naturally defined integrable sub- bHn, defined by equality in (2). The last variable bundles of CT(Cn). Let T 10(Cn) denote the bundle n in (2) plays a different role, and hence for w ∈ C whose sections are vector fields L of the form 0 we write w = (w , wn). Two natural families of n X ∂ biholomorphic self-maps of Hn are the dilations (8) L = aj (z) , 0 ∂zj δr : Hn → Hn, for r > 0, given by δr (w , wn) = j=1 0 2 (rw , r wn), and the rotations RA : Hn → Hn, for where the a are smooth complex-valued functions. 0 0 j A ∈ U(n − 1), given by RA(w , wn) = (A(w ), wn). We note that this bundle is integrable in the sense To introduce an analogue of translation we consider of Frobenius: if K,L are sections of T 10(Cn), then the biholomorphisms τp : Hn → Hn, for p = (ζ, t) ∈ so is their commutator, or Lie bracket, given by n−1 C × R, given by [K, L] = KL − LK. The complex conjugate bundle, 0 0 0 2 denoted T 01(Cn), is also integrable. Since the zero (5) τp(w , wn) = (w +ζ, wn+t+2ihw , ζi+i|ζ| ). section is the only section of both bundles, we All of the preceding maps extend to self-maps of have T 10(Cn) ∩ T 01(Cn) = 0. We obtain a splitting 0 0 2 bHn = {(w , wn) : wn = |w | }. The action of the n 10 n 01 n n−1 CT(C ) = T (C ) ⊕ T (C ). family {τp : p = (ζ, t) ∈ C × R} on Hn ∪ bHn is faithful and the action on bHn is simply transitive. This splitting of the tangent bundle plays a crucial By this method we equip Cn−1 × R with a group role in all aspects of complex geometry. CR law (p, q) , p · q, characterized by the identity geometry studies the extent to which this splitting holds on real manifolds. τp ◦τq = τp·q. The resulting space is the Heisenberg n group. Often in the literature Cn−1 is replaced by Let M be a smooth real hypersurface of C , or R2n−2 and the group law is expressed using real more generally, of a complex manifold. The tangent variables. Finally we observe that the group of spaces of M then inherit some of the ambient complex structure. We write CTM for T (M) ⊗ C. biholomorphic automorphisms of Hn which fix 10 10 n the point at infinity is generated by dilations, We define T (M) = T (C ) ∩ CTM, and denote 01 rotations, and translations. See Stein’s textbook in its complex conjugate bundle by T (M). Again we harmonic analysis [Ste, Chapters XII and XIII] for have the complete story. (9) T 10(M) ∩ T 01(M) = 0.

210 Notices of the AMS Volume 57, Number 2 Let M be an abstract real manifold M with complex- other hand, there is no natural way to resolve the ified tangent bundle CTM. We say that a subbundle ambiguity of sign for an abstract CR manifold. T 10(M) of CTM defines a CR structure on M if it is Next we express the Levi form on a hypersurface integrable and its intersection with its conjugate in terms of partial derivatives. In a neighborhood bundle satisfies (9). We then call M a CR manifold. of a given point we suppose that M is the zero set Its horizontal bundle is the direct sum of r, where r is smooth and dr 6= 0 where r = 0. A (10) H (M) = T 10(M) ⊕ T 01(M). complex vector field L is tangent to M if and only if L(r) = hdr, Li = 0 on M. For the one-form η we We say that M is of hypersurface type if the fibers 1 may use (∂ − ∂)(r). In this case dη = −∂∂r. of H (M) have codimension one in CTM. 2 The Cartan formula for the exterior derivative In general, the CR codimension of M is the of a 1-form η states that codimension of H (M) in CTM. Real submanifolds of Cn of arbitrary dimension define CR manifolds of (11) hdη, A ∧ Bi = Ahη, Bi − Bhη, Ai − hη, [A, B]i. various CR codimensions. Consider a k-dimensional n The commutator term ensures that dη is linear over real subspace of C . As a real submanifold it has the module of smooth functions. Let L and K be codimension 2n − k. On the other hand, its CR local sections of T 10(M). Then hη, Li = hη, Ki = 0. codimension can be any integer ` between 0 and k By the Cartan formula, such that k − ` is even. The subspace has maximal (12) CR codimension (equal to k) if it is totally real, λ(L, K) = hη, [L, K]i = h−dη, L∧Ki = h∂∂r, L∧Ki. and it has minimal CR codimension (equal to zero) if it is complex. Any complex manifold can be We have interpreted the Levi form as the restriction regarded as a CR manifold of CR codimension zero. of the complex Hessian of r to sections of T 10(M). Such manifolds are precisely the integrable almost It is therefore analogous to the real Hessian, which complex manifolds. governs Euclidean convexity. Euclidean convexity We focus on CR manifolds of hypersurface implies pseudoconvexity, but the converse fails. See type. On such manifolds there is a nonvanishing [Hör] and [Kra] for lengthy discussion, especially differential one-form η, defined up to a multiple, for the meaning of pseudoconvexity on a domain annihilating H (M). We may assume that η is whose boundary is not smooth. purely imaginary. Each of the summands in (9) Example 3. The zero set of r(z) = Im(z ) is a half- is integrable, but something new happens; their n H sum is not integrable. The Levi form measures the space Σ. Its horizontal space z (Σ) decomposes failure of integrability of the sum and leads to both into holomorphic and conjugate holomorphic sub- CR geometry and sub-Riemannian geometry. spaces, spanned respectively by the vector fields Lj = ∂/∂zj , j = 1, . . . , n − 1, and their conjugates. Definition 1. Let M be a CR manifold of hypersur- In this case, the complex Hessian of the defining face type. The Levi form λ is the Hermitian form function is identically equal to zero, and hence the on T 10(M) defined by Levi form also vanishes identically.

λ(L, K) = hη, [L, K]i. Pn 2 Example 4. The zero set of r(z) = j=1 |zj | − 1 is 2n−1 2n−1 Any vector field T for which hη, T i 6= 0 has a the sphere S . The horizontal space Hz (S ) 10 component in the missing direction. The Levi form decomposes into the holomorphic subspace Tz 2n−1 λ(L, Li gives the component of [L, L] in the missing (S ) = span{L1,...,Ln−1} and its conjugate 01 2n−1 direction. In the following definition we identify λ Tz (S ), where 10 with a Hermitian linear transformation of T (M). ∂ ∂ (13) Lj = zj − zn . Definition 2. A CR manifold of hypersurface type ∂zn ∂zj is pseudoconvex if all nonzero eigenvalues of λ have The annihilating one-form can be taken to be the same sign. It is called strongly pseudoconvex if n λ is definite, that is, all eigenvalues have the same 1 X (14) η = z dz − z dz . nonzero sign. j j j j 2 j=1 Observe that there is an ambiguity of sign in The Levi form, after dividing out a nonzero fac- the definition of the Levi form. Even for a real tor, satisfies λ(Lj , Lk) = δjk + zj zk. Hence the unit hypersurface in real Euclidean space the sign of the sphere is strongly pseudoconvex. The Heisenberg second fundamental form is defined only modulo group is also strongly pseudoconvex. Its Levi form, the choice of normal. For a compact hypersurface computed similarly, is λ(Lj , Lk) = δjk. in Rn consider a point p farthest from the origin. Since the sphere centered at the origin osculates Remark 5 (Hans Lewy equation). In 1957 Hans M to order two at p, the second fundamental Lewy produced his famous example of an unsolv- form at p agrees with that of the sphere, thus able first-order linear PDE. His equation is Lu = f , determining its sign. The story is the same for where L is the type (1, 0) vector field tangent to S3 real hypersurfaces in Cn and the Levi form. On the given in (13). See [Tre] for an elegant and readable

February 2010 Notices of the AMS 211 account of developments in PDE and CR geometry a differential equation of high order to a first-order based on this operator. system. The references [Mon], [Str], [Gro], [Bel], [FS], [CCG], [CDPT], and [CC] provide many examples Positive definiteness of the second fundamental and discussion of other sub-Riemannian manifolds. form for a real hypersurface M in Rn is a curvature condition: it precludes the existence of straight Differential geometric aspects line segments in M. The next lemma provides a subtle analogue for the Levi form. Let M be a smooth real manifold and let H (M) be a distribution (subbundle) in the tangent bundle Lemma 6. Let M be a strongly pseudoconvex real T (M). The classical Frobenius theorem deals n hypersurface in C . Then M contains no complex with the case when H (M) is integrable, that analytic sets of positive dimension. is, closed under the Lie bracket. In this case M Proof. We prove the contrapositive statement: if M is foliated by H (M)-integral submanifolds. We contains such sets, then M cannot be strongly pseu- consider the opposite extreme, when H (M) is doconvex. Assume M contains a complex analytic completely nonintegrable: the Lie bracket span of set V of positive dimension. Let p be a nonsingular H (M), at any point p ∈ M, coincides with Tp(M). point of V. We can then find a one-dimensional In this case we say that the pair (M, H (M)) satisfies nonsingular holomorphic curve t → z(t) ∈ Cn with the bracket-generating property, also known as z(0) = p, with tangent vector z0(0) 6= 0, and with Hörmander’s condition. Put another way, for each z(t) ⊂ M for |t| < 1. For each local defining func- p ∈ M there exists an integer s = s(p) < ∞ so that tion r for M near p we have r(z(t)) = 0 for |t| the values at p of all s-fold iterated Lie brackets small. of vector fields valued in H (M) fill out the entire ∂ = tangent space T (M). We call s(p) the step of M Taking ∂t of the identity r(z(t)) 0 and evalu- p ating at t = 0 gives at p. We emphasize that s(p) may depend on the point p. In the case when s is uniformly bounded (15) h∂r(p), z0(0)i = 0. on M, we call supp∈M s(p) the step of M. ∂2 Taking the Laplacian ∂t∂t of the same identity and We elaborate with some examples. If M is evaluating at t = 0 gives a CR manifold of hypersurface type, we define 0 H (M) as in (10). The missing direction might or (16) h∂∂r, z (0) ∧ z0(0)i = 0. might not be obtained via iterated Lie brackets 0 10 Equation (15) says that z (0) ∈ Tp (M) and equa- of horizontal vector fields. Example 3 shows that 0 tion (16) says that z (0) is in the null space of Levi flat hypersurfaces do not satisfy the bracket- 0 the Levi form. Since z (0) 6= 0, M is not strongly generating property. The opposite extreme is the pseudoconvex at p. strongly pseudoconvex case. If L is a (1, 0) vector field and not zero at p, then the bracket [L, L] Sub-Riemannian Geometry has a component in the missing direction. In this It has been evident so far that the stratification of case the missing direction arises upon taking a the complexified tangent spaces of a CR manifold single Lie bracket of horizontal vector fields, and given by the horizontal bundle together with the the induced sub-Riemannian structure is of step missing direction is at the heart of CR geometry. two. To be step two we need only one such vector In this section we will consider real manifolds field; we recover the missing direction at p with equipped with a similar horizontal subbundle a single Lie bracket whenever the Levi form is of the (uncomplexified) tangent bundle. In both not zero at p. The simplest example of a higher cases there are horizontal directions which are step structure is the pseudoconvex hypersurface infinitesimally accessible and missing directions. 2m 2 defined by Re(z2) = |z1| in C near the origin. When the Levi form is nonzero we can recover One requires an iterated commutator the missing direction by commutators. The ability to recover missing directions via commutators of [. . . [L, L], . . . , L] horizontal vector fields is the defining property of with 2m total brackets to obtain the missing a sub-Riemannian manifold. This nonintegrability direction. In this case the origin is called a point condition translates to a connectivity condition of type 2m, and one says that the vector field L whereby such manifolds are equipped with a is of type 2m there. Notice in this case that the singular metric. See Theorem 9. step at most points is 2. See page 217 for related CR manifolds provide a natural class of examples, information. but the general theory covers a much wider class of spaces arising in PDE, control theory, geometric Example 7. The Heisenberg group bHn provides group theory, and many other settings. Example the canonical example of a sub-Riemannian mani- 8 puts a sub-Riemannian structure on spaces of fold. The group law was discussed in the paragraph kth order Taylor polynomials (jets). This example following (5). As usual, the left invariant vector formalizes the well-known procedure for reducing fields define the Heisenberg Lie algebra. As a basis

212 Notices of the AMS Volume 57, Number 2 for the horizontal distribution H (bHn) in T (bHn) [0, ∞) (the so-called Carnot-Carathéodory distance): we take the vector fields for p, q ∈ M, d(p, q) is the infimum of (17) Z b ∂ ∂ ∂ ∂ 0 0 1/2 = + = − = − (18) hγ (t), γ (t)i dt Xj 2yj ,Yj 2xj , j 1, . . . , n 1, a ∂xj ∂t ∂yj ∂t over all smooth horizontal curves γ : [a, b] → M = + = = ∂ where wj xj iyj and t Re(wn). Set T ∂t . with γ(a) = p and γ(b) = q. The fundamental Then [Xj ,Yk] = −4T δjk for j, k = 1, . . . , n − 1. By result is Theorem 9 below, the Chow–Rashevsky = 1 − = 1 + putting Lj 2 (Xj iYj ) and Lj 2 (Xj iYj ) and theorem. See the preface in [Mon] for the history regarding them as sections of CT (bHn), we return leading to this theorem. precisely to the CR setting. Theorem 9. Assume that Hörmander’s condition Example 8. Jet spaces provide a geometric inter- is satisfied for the pair (M, H (M)). Then all pairs pretation for Taylor polynomials, by viewing the of points in M are horizontally connected. Conse- equivalence classes of Cm functions modulo m-jets quently, the Carnot-Carathéodory distance function (mth order Taylor approximations) as points in an on a sub-Riemannian manifold M endows M with abstract space. We illustrate in the one-dimensional the structure of a metric space. case. Consider Cm functions f : R → R. The under- lying space M is Rm+2, with coordinates x (cor- An adapted Riemannian metric on M is an extension of the sub-Riemannian metric to the full responding to the base point x0 for the Taylor tangent bundle. If the step of M is finite, then the approximation of f ) and um, . . . , u0 (corresponding (m) topologies defined by the sub-Riemannian metric to f (x0), . . . , f (x0)). The sub-Riemannian structure is defined by a family of smooth 1-forms and any adapted Riemannian metric coincide. However, the geometric and analytic properties ηj = duj − uj+1 dx, j = 0, . . . , m − 1, corresponding to the differential equalities df (j−1)(x) = f (j)(x) dx. of these two metrics differ substantially, unless the step is equal to one. For instance, if M has These forms, together with dx and du , frame the m step at least two, then these two metrics are cotangent bundle T ∗(M). Introduce the dual frame never bi-Lipschitz equivalent. (They are bi-Hölder X,Um,...,U0 for T (M). Let H (M) be spanned by equivalent with Hölder exponent 1 , where s is the X and Um. The Cartan formula (11) gives s step of M.) hηj−1, [Uk,X]i = −hdηj−1,X ∧ Uki Sub-Riemannian geodesics are smooth horizon-

= hduj ∧ dx, Uk ∧ Xi = δjk, tal curves γ in M which locally minimize the length `(γ) defined in (18). The horizontality condition which yields the Lie bracket identities [Uj ,X] = is a family of nonlinear constraints which can be Uj−1, j = 1, . . . , m. We observe that H (M) is a understood in the language of control theory. The bracket-generating horizontal distribution of rank path planning problem for wheeled motion (includ- two inducing a sub-Riemannian structure on M = ing such concrete subproblems as parallel parking m+2 R of step m + 1, called the mth order jet space an automobile) can be reinterpreted as the problem (on the real line). The first-order jet space is iso- of finding geodesics joining prespecified points on morphic (as a Lie group) to the Heisenberg group certain sub-Riemannian manifolds modeled locally bH2. For additional details see [CDPT, Chapter 3.1] on the first Heisenberg group. Again, we reference or [Mon, Chapter 6]. [Mon] for more details. Sub-Riemannian geodesics behave rather dif- The fundamental theorem of sub-Riemannian ferently from their Riemannian counterparts. geometry Abnormal geodesics are locally length minimiz- Recall that a Riemannian metric on a smooth ing curves which fail to satisfy the geodesic manifold M is a smoothly varying family of equations. Such curves cannot exist in Riemannian inner products defined on the tangent spaces. geometry. We refer to [Mon] for a thorough discus- We say that a pair (M, H (M)) satisfying the sion of the geodesics in sub-Riemannian spaces Hörmander condition is a sub-Riemannian manifold and especially the issue of abnormal geodesics. if it is equipped with a smoothly varying family of We remark, however, that all geodesics in the inner products h·, ·ip (the sub-Riemannian metric) sub-Riemannian geometries arising from CR struc- defined on the horizontal tangent spaces Hp(M). tures on the boundaries of strictly pseudoconvex Such a metric permits us to define notions of length, domains are necessarily normal. Additional in- volume, angle and other geometric concepts for formation on the structure of sub-Riemannian objects taking values in the horizontal subbundle. geodesics can be found in [CCG] and the recent For instance, a smooth curve γ : (a, b) → M is publication [CC]. 0 termed horizontal if γ (t) ∈ Hγ(t)M for all t. A rich vein of current research activity in sub- As in the Riemannian case, the sub-Riemannian Riemannian geometry that we do not address is metric induces a distance function d : M × M → the study of the Carnot-Carathéodory geometry of

February 2010 Notices of the AMS 213 submanifolds of sub-Riemannian manifolds. For more information see [CDPT, Chapter 4].

Example 10. We equip the Heisenberg group bHn with a Carnot-Carathéodory metric as follows. The left invariant vector fields from (17) define the horizontal distribution. We declare them to be an orthonormal basis. The geodesics for this metric have a beautiful geometric description. For sim- plicity, we restrict to the case n = 2. In view of the group structure it suffices to describe the geodesics from the identity element o = (0, 0, 0) to an arbitrary point p = (x1, y1, t1) with t1 ≤ 0. Each geodesic is the horizontal lift of a circular arc in the (x, y) plane with endpoints (0, 0) and (x1, y1) so that the area of the planar region bounded by the arc together with the line segment joining these two points is equal to −4t1. The sub-Riemannian distance from o to p is just the Euclidean length of the projected arc. When t1 = 0 the geodesic is just a ray in the t = 0 plane from o to p. When (x1, y1) = (0, 0) geodesics are not unique; the family of geodesics from o to (0, 0, t1) is rotationally symmetric about the t-axis.

The Unit 3 Sphere Just like the circle and the two-sphere, the three-sphere is very round. But there are some beautiful, classical aspects to its roundness Figure 2. Sir William Rowan Hamilton and the that are not easy to guess from its lower- plaque on Dublin’s Brougham Bridge honoring dimensional sisters. Hamilton’s invention of the quaternions. The Mathematics Department of the National William Thurston [Thu] University of Ireland at Maynooth We illustrate the preceding circle of ideas commemorates the occasion with an annual by describing the sub-Riemannian geodesics in walk to the site of this plaque on October 16, 3 the three-sphere S equipped with the Carnot- the anniversary of Hamilton’s discovery. Carathéodory metric arising from its canonical CR structure. These geodesics have recently been computed by several authors using different tech- elementary identities z · j = j · z and z · k = k · z niques. See [HR], [BR], and [CC, Chapter 10]. We for z ∈ C. loosely follow the presentation by Hurtado and We view S3 as the set of unit quaternions Rosales [HR]. The geodesics admit a simple and p = z + z j, where |z |2 + |z |2 = 1. With the elegant description in terms of the Hopf fibration. 1 2 1 2 preceding conventions in place, S3 is equipped with the Lie group law S3 as a Lie group and the Hopf fibration · 0 = + · 0 + 0 Hamilton’s search for an “algebra of vectors in p p (z1 z2j) (z1 z2j) 3 0 0 0 0 R ”, parallel to the identification of plane vectors = (z1z1 − z2z2) + (z1z2 + z1z2)j. with complex numbers, is mathematical folklore. 3 His search led him to develop the theory of the It is well known that S is parallelizable: its tangent quaternions, a noncommutative algebraic structure bundle admits a smoothly varying orthonormal for vectors in R4. An arbitrary quaternion takes frame. The quaternionic presentation provides an the form a + bi + cj + dk, where a, b, c, d ∈ R elegant description for such a frame: just consider = · and i, j, and k are indeterminates satisfying i2 = the vector fields U, V , and W , where U(p) i p, = · = · j2 = k2 = ijk = −1. We will identify quaternions V (p) j p, and W (p) k p. These vector fields with pairs of complex numbers by the rule p = are linked by the bracket relations z + z j ↔ (z , z ) ∈ C2. Thus we identify C with (19) 1 2 1 2 [U, V ] = −2W , [W , U] = −2V , [V , W ] = −2U. the subspace of the quaternions H obtained by setting the coefficients of j and k equal to zero. In particular, setting H (S3) = span{V,W } defines Note that the quaternionic conjugate p is equal to a bracket-generating distribution on S3. If we z1 − z2 · j in this presentation. We point out the set p = z1 + z2 · j, then V (p) = j · (z1 + z2j) =

214 Notices of the AMS Volume 57, Number 2 −z2 + z1j, which coincides with the CR vector field The sub-Riemannian geodesics are critical points L from (13). We have already observed that S3 is for the first variation of (horizontal) length by strongly pseudoconvex, hence step two. Formula horizontal curves with fixed endpoints. Let γ : I → (19) provides the same information. We define a S3 be a C2 horizontal curve defined on a compact sub-Riemannian structure by introducing a family interval I. An admissible variation of γ is a C2 map 3 of inner products on the horizontal bundle for G : I × (−0, 0) → S satisfying which V and W are orthonormal. Our goal is to (1) G(s, 0) = γ(s) for all s ∈ I, describe the geodesics of this sub-Riemannian (2) G(s, ) = G(s, 0) for all and all s ∈ ∂I, structure. (3) for each , γ(s) := G(s, ) is horizontal. The Lie group S3 is isomorphic to the matrix Let X be the vector field along γ whose value at group SU(2) via the identification γ(s) is equal to (∂G/∂)(s, )| . Admissibility of ! =0 z1 z2 the variation can be characterized in terms of X: z1 + z2 · j ←→ . −z2 z1 G is admissible if and only if X vanishes at the endpoints of γ and We can realize SU(2) as the double cover of 0 0 the three-dimensional real rotation group SO(3). (20) γ (hX,Ui) = 2hπH X, J(γ )i, The most elegant description, which dates back to where π denotes projection into the horizontal Cayley, uses the quaternionic presentation: identify H space. R3 with the space of purely imaginary quaternions The classical formula for variation of length in via the basis {i, j, k}, and observe that the map (S3, g), specialized to admissible variations, reads q , p · q · p defines a rotation of R3. In the Z coordinates for the quaternions H described above, ∂ 0 `(γ)|=0 = − hDγ0 γ ,Ui. the map in question takes the form ∂ I ! Considering variations of the form X = f J(γ0) z1 z2 p = ∈ SU(2) , R 1 − p for C functions f : I → R with f |∂I = 0 and z2 z1 R f = 0, and taking into account (20), one finds  2 2  I |z1| − |z2| −2 Im(z1z2) 2 Re(z1z2) that the local minimizers for the sub-Riemannian  2 + 2 − 2 + 2  :=  2 Im(z1z2) Re(z1 z2 ) Im(z1 z2 ) length functional (18) are characterized by the 2 2 2 2 −2 Re(z1z2) Im(z1 + z2 ) Re(z1 + z2 ) Euler–Lagrange equation ∈ SO(3). 0 0 (21) Dγ0 γ + 2κJ(γ ) = 0, This map is two-to-one, as it is invariant under the where κ is a real constant. This constant κ can antipodal map p , −p on S3. be interpreted as a curvature of the geodesic γ. Observe that the first row vector of R lies in p Geodesics of curvature zero are just great circles on S2 for every p. We obtain a map π : S3 → S2. The S3. Geodesics of nonzero curvature are horizontal circle group S1 acts on S3 via the diagonal action lifts of non-great circles on S2 through the Hopf eiθ · (z , z ) = (eiθz , eiθz ), leaving the first row 1 2 1 2 fibration. Indeed, consider a smooth curve γ of the corresponding rotation matrix invariant. taking values in S3 and parameterized by arc Thus we have exhibited S3 as an S1-bundle over S2, length. The tangent space to C2 at p = γ(t) called the Hopf bundle. The induced fibration of S3 splits as T (C2) = T (S3) ⊕ T (S3)⊥, which yields by (geometric) circles is the Hopf fibration. p p p the following decomposition for the acceleration vector: The sub-Riemannian geodesics on S3 00 0 We begin by observing that the annihilating one- (22) γ (t) = Dγ0(t)γ (t) − γ(t). form η from (14) and the corresponding horizontal Then (21) reads γ00 + 2κJ(γ0) + γ = 0. Viewing S3 3 1 distribution H (S ) are invariant under the S as a subset of C2, this equation takes the form action. This remark suggests that the structure of 00 0 the sub-Riemannian geodesics should be related to (23) γ + 2iκγ + γ = 0. the Hopf fibration. We now confirm this suggestion. Equation (23) is a system of constant coefficient Extend the inner product to a Riemannian second-order ODE’s whose explicit solution is easily metric g on S3 by declaring U, V and W to be obtained. Rather than presenting this solution, orthonormal. Let D be the Levi-Civitá connection however, we focus on its geometric content. for this metric g. Define a bundle isomorphism 2 J : H (S3) → H (S3) by setting J(V ) = W and Theorem 11 (Hurtado–Rosales). Let γ be a C hor- 3 J(W ) = −V . The data (S3, H (S3), η, J) defines a izontal curve on S parameterized by arc length. pseudo-Hermitian structure on S3, and the ensuing Then the following are equivalent: discussion can be framed in the language of pseudo- (1) γ is a geodesic of curvature κ in the sub- Hermitian geometry. See, for instance, [CHMY] and Riemannian metric on S3, [DT]. (2) hγ00, J(γ0)i = −2κ,

February 2010 Notices of the AMS 215 (3) γ is the horizontal lift under the Hopf fibra- Group-invariant Mappings between tion S3 → S2 of a piece of a circle in S2 of Spheres in Complex Spaces constant geodesic curvature κ. The unit sphere S3 has dominated our discussion, We recall that the geodesic curvature of a space because of its symmetry and its role in both curve γ contained in S2 is its curvature computed CR and sub-Riemannian geometry. Earlier we 3 relative to the embedding in S2. For a curve γ observed an action of the circle on S defined by iθ iθ parameterized at arbitrary speed, this curvature is (z1, z2) → (e z1, e z2). In this section we study 3 given explicitly by the formula γ · (γ0 × γ00)/|γ0|3. symmetries of S via actions of finite subgroups of U(2). The simplest case is when the group is cyclic; For |h| < 1, the geodesic√ curvature of the circle γ = S2 ∩ {z = h} is h/ 1 − h2. even in the cyclic case there is a remarkable depth We sketch the proof of Theorem 11. To see and breadth of ideas. Given a positive integer p and that (1) and (2) are equivalent, note that the triple an integer q with 1 ≤ q < p, we choose a primitive {γ0, J(γ0), U(γ)} forms an orthonormal frame p-th root of unity α, and consider the group action 3 q for T S along γ. The fact that γ is horizontal (24) (z1, z2) → (αz1, α z2). and arc length parameterized immediately gives 0 The quotient of the sphere by this group is called a that the components of Dγ0 γ in the directions 0 0 lens space, written L(p, q). Lens spaces, introduced γ and U(γ) are equal to zero. Thus D 0 γ is γ by Tietze in 1908, are important in topology. The a multiple of J(γ0) and (21) can be rewritten 0 0 fundamental group of L(p, q) is cyclic of order hDγ0 γ , J(γ )i = −2κ. Equation (22) shows that 00 0 0 0 p for all q, and thus lens spaces with different hγ , J(γ )i = hDγ0 γ , J(γ )i, since γ(t) is normal to the sphere and J(γ0(t)) is tangent to the sphere. p are not homotopy equivalent. Famous work of To see the equivalence of (1) and (3), one calculates Alexander in 1919 showed that the lens spaces the binormal of the Hopf projection of γ, which L(5, 1) and L(5, 2) are not homeomorphic even turns out to be a constant. Thus this projection though they have isomorphic fundamental groups is a circle on S2. Another calculation shows that and the same homology. Necessary and sufficient its geodesic curvature agrees with the curvature κ conditions on p and q for being homeomorphic from (21). are known; similarly, necessary and sufficient con- The topological structure of these geodesics is ditions on p and q for being homotopy equivalent quite interesting. are also known. In Figure 4, the colors represent successive iterations of the group action (24). See Proposition 12 (Hurtado–Rosales). If h := √ κ 1+κ2 http://lukyanenko.net/math/heisenberg09/ is a rational number, then γ is a closed curve, iso- lens.html for animation and additional details. topic to a torus knot inside a group translate of the Clifford torus Th. If h is irrational, then γ is diffeo- morphic to R and is dense in some group translate of Th.

Figure 4. A provocative view of the lens space L(5, 2).

Figure 3. The Hopf fibration and sub-Riemannian geodesics. Lens spaces also arise in CR geometry. We naturally ask the following question. Given a finite subgroup Γ of U(n), is there an integer N and a Curves of the latter type are the well-known nonconstant rational mapping, invariant under Γ , skew lines on the torus. The Clifford tori in S3 from S2n−1 to S2N−1? The answer is, in general, no. In 3 2 are tori of the form Tρ := {(z1, z2) ∈ S : |z1| = fact, according to a theorem of Lichtblau [Lic], such 2 (1 + ρ)/2, |z2| = (1 − ρ)/2}, where |ρ| < 1. Figure a map can exist only when Γ is cyclic. Furthermore, 3 illustrates the Hopf fibration, sub-Riemannian (see [D1]), most representations of cyclic groups geodesics, and the Clifford tori. are also ruled out. We describe the results in the

216 Notices of the AMS Volume 57, Number 2 special case when n = 2. Roughly speaking, the Finite Type and Higher Brackets only possibilities for nonconstant invariant rational Iterated commutators of vector fields provide maps to spheres are the special cases when the a systematic method for dealing with higher quotient space is L(p, 1) or L(p, 2). One amazing order conditions, and their role in sub-Riemannian consequence is that there are homeomorphic lens geometry has been evident in this article. Such spaces, for example L(4, 1) and L(4, 3), only one of conditions are also significant in partial differential which (in this case L(4, 1)) admits a nonconstant equations. Given a finite set of vector fields d rational CR mapping to a sphere. X1,...,Xk in a neighborhood of a point in R , P 2 We do not give up. In order to get more groups consider the operator T = j Xj . Such sums of involved, we can proceed in two ways. First, we can squares of vector fields arise throughout partial weaken the assumptions on the mapping. Smooth differential equations and probability. When the CR mappings in this context must be rational. Sec- Xj form a basis for the tangent space, T is ond, we could allow continuous functions satisfying elliptic, and hence hypoelliptic. Hypoellipticity the tangential Cauchy-Riemann equations in the means that solutions u to the equation T u = f are sense of distributions. We then obtain spheres as smooth when f is smooth. Subellipticity is almost targets of group-invariant maps for any fixed-point as useful as ellipticity, because it also implies free subgroup of U(n). Such results require con- hypoellipticity. An elliptic operator of order two siderable machinery in function theory of several gains two derivatives in the Sobolev scale; a subelliptic operator gains fewer derivatives than its complex variables. If we weaken the assumption order, but the gain suffices to establish regularity on the target by allowing hyperquadrics, then more results. The Hörmander condition, applied to the elementary but fascinating ideas appear. Given a span of the X ’s, provides the necessary and finite subgroup of U(n), by [D1] we can always j Γ sufficient condition for subellipticity of T . find an N and a nonconstant polynomial mapping On the other hand, subellipticity for systems is p : Cn → CN such that p is -invariant, and p maps Γ much more difficult. In two dimensions, subelliptic- the unit sphere to a hyperquadric. The polynomial ity for the Cauchy-Riemann equations reduces to p arises from the following construction. First, one estimates for a scalar sum of squares operator. The defines the Hermitian symmetric polynomial as ΦΓ following theorem precisely relates the gain in a follows: subelliptic estimate with the step of the horizontal Y (z, w) = 1 − (1 − hγz, wi). distribution on the boundary. We omit the technical ΦΓ definition of a subelliptic estimate with gain . γ∈Γ Kohn established the first subelliptic estimate for By elementary linear algebra, there exist holo- weakly pseudoconvex domains in C2 under the morphic vector-valued Γ -invariant polynomial condition of finite step, and Greiner established mappings g, h such that the necessity of this condition. The sharp result N+ N− in Theorem 13 relies on a sophisticated theory X X (z, z) = |g (z)|2 − |h (z)|2. of estimates developed by Rothschild and Stein ΦΓ j j j=1 j=1 [RS] based on the sub-Riemannian geometry of Then the polynomial map p = (g, h) has the nilpotent Lie groups; their theory made it possible desired properties. Even for the cyclic subgroups to relate the geometry and the estimates in a precise fashion. Γ (p, q) used to define lens spaces, the computation of is interesting. For (p, 1) the result is ΦΓ (p,q) Γ Theorem 13. Let Ω be a smoothly bounded pseu- 2 2 p 2 (|z1| + |z2| ) . For Γ (p, 2) and Γ (p, p − 1) explicit doconvex domain in C , and suppose p ∈ bΩ. The formulas exist. For all q, the formula for Γ (p, q) is a following are equivalent: 2 2 polynomial fp,q in the variables |z1| and |z2| with • There is a subelliptic estimate at p with gain integer coefficients. In general these polynomials = 1 2m , but for no larger value of . have many interesting properties; we give here a • For a (1, 0) vector field L on bΩ with L(p) 6= few examples. By [D2], for each q, the congruence 0, we have type(L, p) = 2m. • H = 10 ⊕ 01 f (|z |2, |z |2) |z |2p + |z |2p (mod p) The bundle (M) T (M) T (M) has p,q 1 2 1 2 step 2m at p. holds if and only if p is prime. A combinatorial • The maximum order of contact at p of a interpretation of the integer coefficients appears complex analytic curve with bΩ equals 2m. in [LWW]. See [Mus] for an application to counting In higher dimensions things are more subtle; points on elliptic curves. See [D2] for many addi- algebraic geometry enters the story. Consider the tional properties of these polynomials. See [BR] for hypersurface M in C3 defined by a study of sub-Riemannian geometries on the lens 2 3 2 spaces L(p, q); these geometries are obtained by Re(z3) = |z1 − z2 | . projecting the horizontal distribution from S3 to The step at the origin is 4, and each (1, 0) vector L(p, q) via the natural quotient map. field has type either 4 or 6 there. Thus each such

February 2010 Notices of the AMS 217 vector field is of finite type at the origin. On Acknowledgements the other hand, M contains the complex analytic We would like to thank Harold Parks for encour- 2 3 variety V defined by z1 − z2 = z3 = 0, which has a aging us to write an article for the Notices and for cusp at the origin. (See Figure 5.) At all nonsingular his useful comments on a preliminary draft. We points of V , the step is 2, but there is a (1, 0) vector also acknowledge valuable suggestions from two field (tangent to V ) of infinite type. In particular the referees. The Graduate Program at UIUC provided condition that each (1, 0) vector field be of finite us the opportunity to speak in the Math 499 type holds at the origin but fails at nearby points. series. Luca Capogna, Joe Kohn, Chris Leininger, This failure of openness has consequences in the and Tom Nevins have invariably shared useful study of subelliptic estimates. The condition for geometric insights with us over the years and thus subellipticity must be an open finite-type condition. indirectly aided the preparation of this article. The answer turns out to be that there is a bound on The pictures of lens spaces, the Hopf fibration, the order of contact of (possibly singular) complex and the sub-Riemannian geodesics on S3 were analytic one-dimensional varieties with M at p. See produced by Anton Lukyanenko using Mathemat- [Cat], [CD], [D1], [D3], [DK], and [Koh1]. ica; we gratefully recognize his contribution. We acknowledge the National Science Foundation for research funding under NSF Grants DMS-0500765 and DMS-0753978 (JPD) and Grants DMS-0555869 (JTT).

References [Ahl] L. V. Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable, Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., 1978. [BER] M. S. Baouendi, P. Ebenfelt, and L. P. Roth- schild, Real submanifolds in complex space and their mappings, vol. 47 of Princeton Mathemati- Figure 5. A singular variety in the cal Series, Princeton University Press, Princeton, 3 2 3 2 hypersurface M = {z ∈ C : Re(z3) = |z1 − z2 | }. NJ, 1999. [Bel] S. R. Bell, Biholomorphic mappings and the ∂- problem, Annals of Math. 114 (1981), 103–113. [BR] U. Boscain and F. Rossi, Invariant Carnot- Carathéodory metrics on S3, SO(3), SL(2) and Concluding Remarks lens spaces, SIAM J. Control Optim. 47 (2008), Math 499, a required course for first-year graduate 1851–78. students at the University of Illinois, exposes [CC] O. Calin and D.-C. Chang, Sub-Riemannian students to a wide variety of mathematical points of Geometry: General Theory and Examples, view through engaging invitations to the research Cambridge University Press, Cambridge, UK, areas of the speakers. This note arose from 2009. [CCG] O. Calin, D.-C. Chang and P. Greiner, Geometric discussing the background common to talks we analysis on the Heisenberg group and its gener- have given in this course but includes material well alizations, vol. 40 of AMS/IP Studies in Advanced beyond what we actually presented. We hope that Mathematics, American Mathematical Society, we have provided sufficient insights and references Providence, RI; International Press, Somerville, to motivate further study. MA, 2007. Much of the material here is only tangentially [CDPT] L. Capogna, D. Danielli, S. D. Pauls, and J. T. related to differential geometry, especially if one Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, appeals to the hilarious definition “Differential vol. 259 of Progress in Mathematics, Birkhäuser, geometry is the study of those properties invariant Basel, 2007. under change of notation” sometimes attributed [Cat] D. W. Catlin, Subelliptic estimates for the ∂- to Calabi. Regardless of the definition, differential Neumann problem on pseudoconvex domains, geometry encompasses a diverse collection of Annals of Math. 126 (1987), 131–91. subjects, including both CR geometry and sub- [CD] D. W. Catlin and J. P. D’Angelo, Subelliptic Riemannian geometry. While both these subjects estimates, to appear in Complex Analysis: Sev- eral complex variables and connections with PDEs have their roots in mapping theorems from complex and geometry (Fribourg 2008), Trends in Math, analysis, they have diverged considerably over the Birkhäuser-Verlag. past forty years. Perhaps this article is a small step [CNS] D.-C. Chang, A. Nagel, and E. M. Stein, toward their eventual convergence. Estimates for the ∂-Neumann problem in

218 Notices of the AMS Volume 57, Number 2 pseudoconvex domains of finite type in C2, [Lic] D. Lichtblau, Invariant proper holomorphic Acta Math. 169 (1992), 153–228. maps between balls, Indiana Univ. Math. J. 41(1) [CHMY] J.-H. Cheng, J.-F. Hwang, A. Malchiodi, and (1992), 213–31. P. Yang, Minimal surfaces in pseudohermitian [LWW] N. A. Loehr, G. S. Warrington, and H. S. geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Wilf, The combinatorics of a three-line circulant 4(1) (2005), 129–77. determinant, Israel J. Math. 143 (2004), 141–56. [Cho] W. L. Chow, Über Systeme von linearen partiellen [Mon] R. Montgomery, A tour of sub-Riemannian ge- Differentialgleichungen erster Ordnung, Math. ometries, their geodesics and applications, vol. 91 Ann. 117 (1939), 98–105. of Mathematical Surveys and Monographs, Amer. [D1] J. P. D’Angelo, Several complex variables and Math. Soc., Providence, RI, 2002. the geometry of real hypersurfaces, in Studies in [Mus] G. Musiker, Combinatorial aspects of elliptic Advanced Mathematics, CRC Press, Boca Raton, curves, Séminaire Lotharingien de Combinatoire FL, 1993. 56 (2007), Article B56f. [D2] , Number-theoretic properties of certain [RS] L. P. Rothschild and E. M. Stein, Hypoelliptic CR mappings, J. Geom. Anal. 14(2) (2004), 215– differential operators and nilpotent groups, Acta 29. Math. 137 (1976), 247–320. [D3] , Real and complex geometry meet the [Ste] E. M. Stein, Harmonic analysis: real-variable Cauchy-Riemann equations, Park City Math methods, orthogonality, and oscillatory inte- Institute Notes, 2008 (to appear). grals, vol. 43 of Princeton Mathematical Series, [DK] J. P. D’Angelo and J. J. Kohn, Subelliptic esti- Princeton University Press, Princeton, NJ, 1993, mates and finite type, pp. 199–232 in Several with the assistance of Timothy S. Murphy, Complex Variables (M. Schneider and Y. T. Siu, Monographs in Harmonic Analysis, III. eds.), Math Sci. Res. Inst. Publ., vol. 37, Cambridge [Str] R. S. Strichartz, Sub-Riemannian geometry, J. Univ. Press, 1999. Differential Geom. 24(2) (1986), 221–63. [DT] S. Dragomir and G. Tomassini, Differential [Thu] W. P. Thurston, Three-dimensional geometry geometry and analysis on CR manifolds, vol- and topology, Vol. 1, edited by Silvio Levy, ume 246 of Progress in Mathematics, Birkhäuser Princeton Mathematical Series, vol. 35, Princeton Boston Inc., Boston, MA, 2006. University Press, Princeton, NJ, 1997. [Fef] Ch. Fefferman, The Bergman kernel and biholo- [Tre] F. Treves, A treasure trove of geometry and anal- morphic mappings of pseudoconvex domains, ysis: The hyperquadric, Notices Amer. Math. Soc. Invent. Math. 26 (1974), 1–65. 47(10) (2000), 1246–56. [FS] G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. [Fri] B. L. Fridman, An approximate Riemann mapping theorem in Cn, Math. Ann. 275(1) (1986), 49–55. [Gro] M. Gromov, Carnot-Carathéodory spaces seen from within, in Sub-Riemannian geometry, 79–323, Progress in Mathematics, vol. 144, Birkhäuser, Basel, 1996. [Hör] L. Hörmander, An introduction to complex analysis in several variables, vol. 7 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, third edition, 1990. [HR] A. Hurtado and C. Rosales, Area-stationary surfaces inside the sub-Riemannian three-sphere, Math. Ann. 340(3) (2008), 675–708. [Jac] H. Jacobowitz, Real hypersurfaces and complex analysis, Notices Amer. Math. Soc. 42(12) (1995), 1480–8. [Koh1] J. J. Kohn, Subellipticity of the ∂¯-Neumann problem on pseudoconvex domains: sufficient conditions, Acta Math. 142 (1979). [Koh2] , Boundary behavior of ∂¯ on weakly pseudoconvex manifolds of dimension two, in A collection of articles dedicated to S. S. Chern and D. C. Spencer on their sixtieth birthdays, J. Diff. Geom. 6 (1972), 523–42. [Kra] S. G. Krantz, Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1992 edition.

February 2010 Notices of the AMS 219