On the Dimension of the Stability Group for a Levi Non-Degenerate Hypersurface

Transcription

1 1 On the Dimension of the Stability Group for a Levi Non-Degenerate Hypersurface Vladimir Ezhov and Alexander Isaev We classify locally defined non-spherical real-analytic hypersurfaces in complex space whose Levi form has no more than one negative eigenvalue and for which the dimension of the group of local CR-automorphisms has the second largest value. 1 Introduction Let M be a real-analytic hypersurface in C n+1 passing through the origin. Assume that the Levi form of M at 0 is non-degenerate and has signature (n m, m) with n 2m. Then in some local holomorphic coordinates z = (z 1,..., z n ), w = u + iv in a neighborhood of the origin, M can be written in the Chern-Moser normal form (see [CM]), that is, given by an equation v = z, z + F kl (z, z, u), where z, z = n α,β=1 k,l 2 h αβ z α z β is a non-degenerate Hermitian form with signature (n m, m), and F kl (z, z, u) are polynomials of degree k in z and l in z whose coefficients are analytic functions of u such that the following conditions hold tr F 22 0, tr 2 F 23 0, (1.1) tr 3 F Mathematics Subject Classification: 32V40 Keywords and Phrases: Levi non-degenerate hypersurfaces, Chern-Moser normal forms, linearization of local CR-automorphisms.

2 2 V. Ezhov and A. Isaev Here the operator tr is defined as tr := n α,β=1 ĥ αβ 2 z α z β, where (ĥαβ) is the matrix inverse to H := (h αβ ). Everywhere below we assume that M is given in the normal form. Let Aut 0 (M) denote the group of all local CR-automorphisms of M defined near 0 and preserving 0. To avoid confusion with the term isotropy group of M at 0 usually reserved for global CR-automorphisms of M preserving the origin, this group is often called the stability group of M at 0. Every element ϕ of Aut 0 (M) extends to a biholomorphic mapping defined in a neighborhood of the origin in C n+1 and therefore can be written as z f ϕ (z, w), w g ϕ (z, w), where f ϕ and g ϕ are holomorphic. We equip Aut 0 (M) with the topology of uniform convergence of the partial derivatives of all orders of the component functions on a neighborhood of 0. The group Aut 0 (M) with this topology is a topological group. It follows from [CM] that every element ϕ = (f ϕ, g ϕ ) of Aut 0 (M) is uniquely determined by a set of parameters (U ϕ, a ϕ, λ ϕ, σ ϕ, r ϕ ), where σ ϕ = ±1, U ϕ is an n n-matrix such that U ϕ z, U ϕ z = σ ϕ z, z for all z C n, a ϕ C n, λ ϕ > 0, r ϕ R (note that σ ϕ can be equal to 1 only for n = 2m). These parameters are determined by the following relations f ϕ z (0) = λ ϕu ϕ, f ϕ w (0) = λ ϕu ϕ a ϕ, g ϕ w (0) = σ ϕλ 2 ϕ, Re 2 g ϕ 2 w (0) = 2σ ϕλ 2 ϕr ϕ. For results on the dependence of local CR-mappings on their jets in more general settings see [BER1], [BER2], [Eb], [Z]. We assume that M is non-spherical at the origin, i.e., that M in a neighborhood of the origin is not CR-equivalent to an open subset of the hyperquadric given by the equation v = z, z. In this case for every element

3 Dimension of the Stability Group 3 ϕ = (f ϕ, g ϕ ) of Aut 0 (M) the parameters a ϕ, λ ϕ, σ ϕ, r ϕ are uniquely determined by the matrix U ϕ, and the mapping Φ : Aut 0 (M) GL n (C), Φ : ϕ U ϕ is a continuous injective homomorphism of topological groups whose range G 0 := Φ(Aut 0 (M)) is a real algebraic subgroup of GL n (C); in addition the mapping Λ : G 0 (M) R +, Λ : U ϕ λ ϕ (1.2) is a Lie group homomorphism with the property Λ(U ϕ ) = 1 if all eigenvalues of U ϕ are unimodular, where R + is the group of positive real numbers with respect to multiplication (see [B], [L1], [BV], [VK]). Since G 0 (M) is a closed subgroup of GL n (C), we can pull back its Lie group structure to Aut 0 (M) by means of Φ (note that the pulled back topology may a priori be different from that of Aut 0 (M), but no such examples are known). We are interested in the dimension d 0 (M) of Aut 0 (M) with this Lie group structure. If n > 2m, G 0 (M) is a closed subgroup of the pseudounitary group U(n m, m) of all matrices U such that U t HU = H, where H is the matrix of the Hermitian form z, z. The group U(n, 0) is the unitary group U(n). If n = 2m, G 0 is a closed subgroup of the group U (m, m) of all matrices U such that U t HU = ±H, that has two connected components. In particular, we always have d 0 (M) n 2. If d 0 (M) = n 2 and n > 2m, then G 0 (M) = U(n m, m). If d 0 (M) = n 2 and n = 2m, then we have either G 0 (M) = U(m, m), or G 0 (M) = U (m, m). Observe that if d 0 (M) = n 2, the mapping Λ defined in (1.2) is constant, that is, λ ϕ = 1 for all ϕ Aut 0 (M). Indeed, consider the restriction of Λ to U(n m, m). Every element U U(n m, m) can be represented as U = e iψ V with ψ R and V SU(n m, m). Note that there are no nontrivial homomorphisms from the unit circle into R + since R + has no nontrivial compact subgroups. Also, there are no non-trivial homomorphisms from SU(n m, m) into R + since the kernel of any such homomorphism is a proper normal subgroup of SU(n m, m) of positive dimension, and

4 4 V. Ezhov and A. Isaev SU(n m, m) is a simple group. Thus, Λ is constant on U(n m, m) and hence on all of G 0 (M). We will say that the group Aut 0 (M) is linearizable, if in some coordinates (that can always be chosen to be normal) every ϕ Aut 0 (M) can be written in the form z λu z, w σλ 2 (1.3) w. Clearly, in the above formula U = U ϕ, λ = λ ϕ, σ = σ ϕ. If Aut 0 (M) is linearizable, then Φ is a homeomorphism and Aut 0 (M) is a Lie group isomorphic to G 0 (M) in the original topology of Aut 0 (M). The group Aut 0 (M) is known to be linearizable for m = 0 (see [KL]) and for m = 1 (see [Ezh1]). Suppose that Aut 0 (M) is linearizable and d 0 (M) = n 2. Choose local holomorphic coordinates near the origin in which every element of Aut 0 (M) has the form (1.3). Then, since Λ in this case is a constant mapping, the function F (z, z, u) := F kl (z, z, u) k,l 2 is invariant under all linear transformations of the z-variables from U(n m, m) and therefore depends only on z, z and u. Conditions (1.1) imply that F 22 0, F Thus, F has the form F (z, z, u) = C k (u) z, z k, (1.4) k=4 where C k (u) are real-valued analytic functions of u, and for some k we have C k (u) 0. Note, in particular, that if d 0 (M) = n 2, then 0 is an umbilic point for M. Conversely, if M is given by an equation v = z, z + F (z, z, u), with F 0 of the form (1.4), then Aut 0 (M) contains all linear transformations (1.3) with U U(n m, m), λ = 1 and σ = 1, and therefore d 0 (M) = n 2. For n > 2m and for n = 2m with G 0 (M) = U(m, m), Aut 0 (M) clearly coincides with the group of all transformations of the form z Uz, w w. (1.5)

5 Dimension of the Stability Group 5 where U U(n m, m). If n = 2m and G 0 (M) = U (m, m), then Aut 0 (M) consists of all mappings z Uz, w σw, where U U (m, m), Uz, Uz = σ z, z, σ = ±1 (note that by [L2] all elements of Aut 0 (M) are linear transformations). We are interested in characterizing hypersurfaces for which m is either 0 or 1 with d 0 (M) being strictly less than the maximal dimension n 2. From now on we assume that M is given in normal coordinates where every ϕ Aut 0 (M) is a linear mappings of the form (1.3). For the strongly pseudoconvex case we obtain the following THEOREM 1.1 Let M be a strongly pseudoconvex real-analytic non-spherical hypersurface in C n+1 with n 2 (here m = 0). Then the following holds (i) d 0 (M) n 2 2n + 3 implies d 0 (M) = n 2 ; (ii) d 0 (M) = n 2 2n+2 if and only if after a linear change of the z-coordinates the equation of M takes the form n v = z α 2 + F (z, z, u), (1.6) α=1 n where F is a function of z 1 2, z, z := z α 2 and u: α=1 F (z, z, u) = p+q 4 C pq (u) z 1 2p z, z q, (1.7) where C pq (u) are real-valued analytic functions of u, and C pq (u) 0 for some p, q with p > 0. In these coordinates the group Aut 0 (M) coincides with the group of all mappings of the form (1.5), where U U(1) U(n 1) (with U(1) U(n 1) realized as a group of block-diagonal matrices in the standard way). Corollary 1.2 If M is a strongly pseudoconvex real-analytic hypersurface in C n+1 with n 2, and the dimension of Aut 0 (M) is greater than or equal to n 2 2n + 2, then the origin is an umbilic point for M.

6 6 V. Ezhov and A. Isaev For the case m = 1 we prove the following THEOREM 1.3 Let M be a Levi non-degenerate real-analytic non-spherical hypersurface in C n+1 with m = 1. Then the following holds (i) d 0 (M) n 2 2n + 4 implies d 0 (M) = n 2 ; (ii) d 0 (M) = n 2 2n+3 if and only if after a linear change of the z-coordinates the equation of M takes the form v = 2Re z 1 z n + n 1 α=2 where F is a function of z n 2, z, z := 2Re z 1 z n + z α 2 + F (z, z, u), (1.8) n 1 α=2 z α 2 and u: F (z, z, u) = C rpq u r z n 2p z, z q, (1.9) where at least one of C rpq R is non-zero, the summation is taken over p 1, q 0, r 0 such that (r + q 1)/p = s with s 1/2 being a fixed rational number, and F (z, z, u) = k,l 2 F kl (z, z, u), where F 23 = 0 and identities (1.1) hold for F 22 and F 33. In these coordinates the group Aut 0 (M) coincides with the group of all mappings of the form z a 1/(s+1) Uz, w a 2/(s+1) (1.10) w, with U S, where S is the group introduced in Lemma 3.1 below, and a is a parameter in this group (see formula (3.2)). Corollary 1.4 Let M be a Levi non-degenerate real-analytic hypersurface in C n+1 with m = 1, and assume that the dimension of Aut 0 (M) is greater

7 Dimension of the Stability Group 7 than or equal to n 2 2n + 3. If the origin is a non-umbilic point for M, then in some normal coordinates the equation of M takes the form v = 2Re z 1 z n + n 1 α=2 z α 2 ± z n 4. (1.11) We remark that hypersurfaces (1.11) occur in [P] in connection with studying unbounded homogeneous domains in complex space. The proofs of Theorems 1.1 and 1.3 are given in Sections 2 and 3 respectively. It would be interesting to extend these proofs to cases for which Aut 0 (M) is not known to be linearizable. Before proceeding we wish to acknowledge that this work was done while the first author was visiting the Mathematical Sciences Institute of the Australian National University. 2 The Strongly Pseudoconvex Case First of all, we note that the mapping Λ defined in (1.2) is constant, that is, λ ϕ = 1 for all ϕ Aut 0 (M). This follows from the fact that all eigenvalues of U ϕ are unimodular, or, alternatively, from the compactness of G 0 (M) and the observation that R + does not have non-trivial compact subgroups. Next, by a linear change of the z-coordinates the matrix H can be transformed into the identity matrix E, and for the remainder of this section we assume that H = E. Hence the equation of M is written in the form (1.6), where the function F satisfies the normal form conditions. It is shown in Lemma 2.1 of [IK] that any closed subgroup of the unitary group U(n) of dimension n 2 2n + 3 or larger is either SU(n) or U(n) itself. Hence, if d 0 (M) n 2 2n + 3, we have G 0 (M) SU(n), and therefore F (z, z, u) is invariant under all linear transformations of the z-variables from SU(n). This implies that F (z, z, u) is a function of z, z and u, which gives that F (z, z, u) is invariant under the action of the full unitary group U(n) and thus d 0 (M) = n 2, as stated in (i). The proof of part (ii) of the theorem is also based on Lemma 2.1 of [IK]. For the case d 0 (M) = n 2 2n + 2 the lemma gives that G 0 is either conjugate in U(n) to the subgroup U(1) U(n 1) realized as block-diagonal matrices, or, for n = 4, contains a subgroup conjugate to Sp 2,0. If the latter

8 8 V. Ezhov and A. Isaev is the case, then, since Sp 2,0 acts transitively on the sphere of dimension 7 in C 4, F (z, z, u) is a function of z, z and u, which implies that F (z, z, u) is invariant under the action of the full unitary group U(4) and thus d 0 (M) = 16, which is impossible. Hence G 0 is conjugate to U(1) U(n 1), and therefore, after a unitary change of the z-coordinates, the equation of M can be written in the form (1.6) where the function F depends on z 1 2, n z, z := z α 2 and u. Clearly, z, z = z, z z 1 2, and F can be α=2 written as a function of z 1 2, z, z and u as in (1.7). Next, conditions (1.1) imply that F 22 0, F 33 0, and thus the summation in (1.7) is taken over p, q such that p + q 4. Further, if C pq 0 for all p > 0, F has the form (1.4) and therefore G 0 = U(n) which is impossible. Thus for some p, q with p > 0 we have C pq 0. Conversely, if M is written in the form (1.6) with function F as in (1.7), it follows from [L2] that every element of Aut 0 (M) has the form (1.5). Clearly, G 0 (M) contains U(1) U(n 1). Hence d 0 (M) n 2 2n+2. If d 0 (M) > n 2 2n+2, then by part (i) of the theorem, d 0 (M) = n 2 and hence G 0 (M) = U(n). Then F has the form (1.4) which is impossible because for some p, q with p > 0 the function C pq does not vanish identically. Thus d 0 (M) = n 2 2n+2, and Lemma 2.1 of [IK] gives that G 0 (M) = U(1) U(n 1). Therefore Aut 0 (M) coincides with the group of all mappings of the form (1.5), where U U(1) U(n 1). Thus, (ii) is established, and the theorem is proved. 3 The Case of U(n 1, 1) We start with the following algebraic lemma. Lemma 3.1 Let G U(n 1, 1) be a real algebraic subgroup of GL n (C), with Hermitian form preserved by U(n 1, 1) written as E , (3.1)

9 Dimension of the Stability Group 9 where E is the (n 2) (n 2) identity matrix. Then the following holds (a) if dim G n 2 2n+4, we have either G = SU(n 1, 1), or G = U(n 1, 1); (b) if dim G = n 2 2n + 3, the group G is conjugate in U(n 1, 1) to the group S that consists of all matrices of the form a ax T A c 0 A x, (3.2) 0 0 1/a where a, c C, a 0, x C n 2, A U(n 2) (i.e., A is an (n 2) (n 2)- matrix with complex elements such that A T A = E), and the following holds 2Re c a + xt x = 0. Proof: Let V U(n 1, 1) be a real algebraic subgroup of GL n (C) such that dim V n 2 2n + 3. Consider V 1 := V SU(n 1, 1). Clearly, dim V 1 n 2 2n + 2. Let V1 C SL n (C) be the complexification of V 1. We have dim C V1 C n 2 2n+2. Consider the maximal complex closed subgroup W (V ) SL n (C) that contains V1 C. Clearly, dim C W (V ) n 2 2n + 2. All closed maximal subgroups of SL n (C) had been classified (see [D]), and the lower bound on the dimension of W (V ) gives that either W (V ) = SL n (C), or W (V ) is conjugate to one of the parabolic subgroups P 1 := P 2 := {( 1/ det C b 0 C {( C b 0 1/ det C ) ), b C n 1, C GL n 1 (C), b C n 1, C GL n 1 (C) (note that P 1 = P 2 for n = 2), or, for n = 4, W (V ) is conjugate to Sp 4 (C). Suppose that for some g SL n (C) and j we have g 1 W (V )g = P j. It is not hard to show that, due to the lower bound on the dimension of W (V ), g can be chosen to belong to SU(n 1, 1). Then g 1 V 1 g P j SU(n 1, 1). } },

10 10 V. Ezhov and A. Isaev It is easy to compute the intersections P j SU(n 1, 1) for j = 1, 2 and see that they are equal and coincide with the group S 1 of matrices of the form (3.2) with determinant 1. Clearly, dim S 1 = n 2 2n + 2 dim V 1 and therefore V 1 is conjugate to S 1 in SU(n 1, 1). Suppose now that n = 4 and for some g SL n (C) we have g 1 W (V )g = Sp 4 (C). In particular, g 1 V 1 g Sp 4 (C) g 1 SU(n 1, 1)g. It can be shown that dim Sp 4 (C) g 1 SU(n 1, 1)g 6 for all g SL n (C). At the same time we have dim V This contradiction shows that W (V ) in fact cannot be conjugate to Sp 4 (C). Suppose now that dim G n 2 2n+4. Then dim G 1 n 2 2n+3, and the above considerations give that W (G) = SL n (C). Hence G 1 = SU(n 1, 1) which implies that either G = SU(n 1, 1), or G = U(n 1, 1), thus proving (a). Let dim G = n 2 2n+3. In this case we can only have dim G 1 = n 2 2n+2, which implies that G 1 is conjugate to S 1 in SU(n 1, 1). Therefore, G is conjugate to S in U(n 1, 1), and (b) is established. The lemma is proved. We will now prove Theorem 1.3. By a linear change of the z-coordinates the matrix H can be transformed into matrix (3.1), and from now on we assume that H is given in this form. Hence the equation of M is written as in (1.8), where the function F satisfies the normal form conditions. Lemma 3.1 gives that if d 0 (M) n 2 2n+4, then we have either G 0 (M) = SU(n 1, 1), or G 0 (M) = U(n 1, 1), or, for n = 2, G 0 (M) = U (1, 1). In each of these cases there are no non-trivial homomorphisms from G 0 (M) into R +, and thus the mapping Λ defined in (1.2) is constant, that is, λ ϕ = 1 for all ϕ Aut 0 (M). Therefore F (z, z, u) is invariant under all linear transformations of the z-variables from SU(n 1, 1), which implies, as in the proof of Theorem 1.1, that d 0 (M) = n 2, and (i) is established. Suppose now that d 0 (M) = n 2 2n + 3. In this case Lemma 3.1 implies that after a linear change of the z-coordinates preserving the form H the following holds: for every U S (where S is the group defined in (3.2)) the equation of M is invariant under the linear transformation z λ U Uz, w λ 2 Uw, (3.3)

11 Dimension of the Stability Group 11 where λ U = Λ(U). The group S contains U(n 2) realized as the subgroup of all matrices of the form (3.2) with a = 1, c = 0, x = 0. Since Λ is constant on U(n 2), we have λ U = 1 for all U U(n 2). Therefore, n 2 the function F (z, z, u) depends on z 1, z n, z 1, z n, z, z := z α 2 and u. Clearly, z, z = z, z 2Re z 1 z n, and F can be written as follows F (z, z, u) = D rq (z 1, z n, z 1, z n )u r z, z q, r,q 0 where D rq are real-analytic. We will now determine the form of the functions D rq. The group S contains the subgroup I of all matrices as in (3.2) with a = 1, x = 0 and A = E. Since every eigenvalue of any U I has absolute value 1, we have λ U = 1 for all U I, and therefore D rq is invariant under all linear transformations from I. It then follows from [Ezh2] that D rq is a function of Re z 1 z n and z n 2. Let further J be the subgroup of S given by the conditions a = 1, A = E. For every U J we also have λ U = 1, and hence D rq is invariant under all linear transformations from J. It is then easy to see that D rq has to be a function of z n 2 alone. Thus, the function F has the form (1.9), and it remains to show that the summation in (1.9) is taken over p 1, q 0, r 0 such that (r + q 1)/p = s, where s 1/2 is a fixed rational number. Let K be the 1-dimensional subgroup of S given by the conditions a > 0, c = 0, x = 0, A = E. It is straightforward to show that every homomorphism Ψ : K R + has the form U a α, where α R. Considering Ψ = Λ K we obtain that there exists α R such that for every U K we have λ U = a α. We will now prove that α 0. Indeed, otherwise F would be invariant under all linear transformations from K and therefore would be a function of z, z and u, which implies that G 0 (M) = U(n 1, 1). This contradiction shows that α 0 and hence λ U 1 for every U K with a 1. Plugging a mapping of the form (3.3) with U K, a 1, into equation (1.8), where F 0 has the form (1.9) we obtain that, if C rpq 0, then α=2 λ r+p+q 1 U = a p. (3.4) The equation of M is written in the normal form, hence p + q 2 and r + p + q 1 1. Since λ U 1, we obtain that p 1. Further (3.4) implies λ (r+p+q 1)/p U = a,

12 12 V. Ezhov and A. Isaev and, since the right-hand side in the above identity does not depend on r, p, q, for all non-zero coefficients C rpq the ratio (r + q 1)/p must have the same value; we denote it by s. Clearly, s is a rational number and s 1/2. We also remark that α = p/(r + p + q 1) = 1/(s + 1). Conversely, suppose that equation (1.8) of M is given in the normal form with F 0 as in (1.9), and the summation in (1.9) is taken over p 1, q 0, r 0 such that (r + q 1)/p = s, where s 1/2 is a fixed rational number. It follows from [L2] that every element of Aut 0 (M) has the form (1.3). Set α = 1/(s + 1) and for every U S define λ U = a α. It is then straightforward to verify that every mapping of the form (3.3) with U S is an automorphism of M. Therefore, G 0 (M) contains S and hence d 0 (M) n 2 2n + 3. If d 0 (M) > n 2 2n + 3, then by part (i) of the theorem, d 0 (M) = n 2 and hence G 0 (M) = U(n 1, 1). Then F has the form (1.4) which is impossible since for every non-zero C rpq we have p 1. Hence d 0 (M) = n 2 2n + 3. If n > 2, Lemma 3.1 gives that G 0 (M) = S, and therefore Aut 0 (M) coincides with the group of all mappings of the form (1.10). If n = 2, it is a priori possible that G 0 (M) contains elements from the second connected component of the group U (1, 1). This component is equal to g 0 U(1, 1), where ( ) 0 1 g 0 =. 1 0 It is straightforward to verify, however, that no transformation of the form z λu z, w λ 2 w with U g 0 U(1, 1) and λ > 0 preserves equation (1.8) with F as in (1.9). Therefore Aut 0 (M) coincides with the group of all mappings of the form (1.10) for n = 2 as well. Thus, (ii) is established, and the theorem is proved. References [BER1] Baouendi, M. S., Ebenfelt, P., Rothschild, L. P., Rational dependence of smooth and analytic CR mappings on their jets, Math. Ann. 315(1999),

13 Dimension of the Stability Group 13 [BER2] [B] [BV] [CM] [D] [Eb] [Ezh1] [Ezh2] [IK] [KL] [L1] Baouendi, M. S., Ebenfelt, P., Rothschild, L. P., CR automorphisms of real analytic manifolds in complex space, Comm. Anal. Geom. 6(1998), Beloshapka, V. K., On the dimension of the group of automorphisms of an analytic hypersurface (translated from Russian), Math. USSR- Izv. 14(1980), Beloshapka, V.K. and Vitushkin, A.G., Estimates for the radius of convergence of power series defining mappings of analytic hypersurfaces (translated from Russian), Math. USSR-Izv. 19(1982), Chern, S. S. and Moser, J. K., Real hypersurfaces in complex manifolds, Acta Math. 133(1974), Dynkin, E. B., Maximal subgroups of the classical groups (translated from Russian), Amer. Math. Soc. Translations II, Ser. 6 (1957), Ebenfelt, P., Finite jet determination of holomorphic mappings at the boundary, Asian J. Math. 5(2001), Ezhov, V. V., Linearization of stability group for a class of hypersurfaces (translated from Russian), Russian Math. Surveys 41(1986), Ezhov, V.V., On the linearization of automorphisms of a real analytic hypersurface (translated from Russian), Math. USSR-Izv. 27(1986), Isaev, A. V. and Krantz, S. G., On the automorphism groups of hyperbolic manifolds, J. Reine Angew. Math. 534(2001), Kruzhilin, N. G. and Loboda, A.B., Linearization of local automorphisms of pseudoconvex surfaces (translated from Russian), Sov. Math., Dokl. 28(1983), Loboda, A. V., On local automorphisms of real-analytic hypersurfaces (translated from Russian), Math. USSR-Izv. 20(1983),

14 14 V. Ezhov and A. Isaev [L2] [P] [VK] [Z] Loboda, A.V., Linearizability of automorphisms of non-spherical surfaces (translated from Russian), Math. USSR-Izv. 21(1983), Penney, R., Homogeneous Koszul manifolds in C n, J. Diff. Geom. 36(1992), Vitushkin, A. G. and Kruzhilin, N. G., Description of the automorphism groups of real hypersurfaces of complex space (Russian), Investigations in the theory of the approximation of functions, Akad. Nauk SSSR Bashkir. Filial, Otdel Fiz. Mat., Ufa, 1987, Zaitsev, D., Unique determination of local CR-maps by their jets: a survey, in Harmonic analysis on complex homogeneous domains and Lie groups (Rome, 2001), Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 13(2002), School of Mathematics and Statistics University of South Australia Mawson Lakes Blvd Mawson Lakes South Australia 5091 AUSTRALIA vladimir.ejov@unisa.edu.au Department of Mathematics The Australian National University Canberra, ACT 0200 AUSTRALIA alexander.isaev@maths.anu.edu.au