164. On the CR automorphism group of a certain hypersurface of infinite type in ℂ 2

This article was downloaded by: [University of Otago] On: 26 December 2014, At: 19:36 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Complex Variables and Elliptic Equations: An International Journal Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gcov20 On the CR automorphism group of a certain hypersurface of infinite type in ℂ a Ninh Van Thu a Department of Mathematics, Vietnam National University, 334 Nguyen Trai Str., Hanoi, Vietnam Published online: 24 Dec 2014 Click for updates To cite this article: Ninh Van Thu (2014): On the CR automorphism group of a certain hypersurface of infinite type in ℂ , Complex Variables and Elliptic Equations: An International Journal, DOI: 10.1080/17476933.2014.986656 To link to this article: http://dx.doi.org/10.1080/17476933.2014.986656 PLEASE SCROLL 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or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden Terms & Downloaded by [University of Otago] at 19:36 26 December 2014 Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions Complex Variables and Elliptic Equations, 2014 http://dx.doi.org/10.1080/17476933.2014.986656 On the CR automorphism group of a certain hypersurface of infinite type in C2 Ninh Van Thu∗1 Department of Mathematics, Vietnam National University, 334 Nguyen Trai Str., Hanoi, Vietnam Communicated by S Ivashkovich Downloaded by [University of Otago] at 19:36 26 December 2014 (Received September 2014; accepted November 2014) We consider C ∞ -smooth real hypersurfaces of infinite type in C2 The purpose of this paper is to give explicit descriptions for stability groups of the hypersurface M(a, α, p, q) and a radially symmetric hypersurface in C2 Keywords: holomorphic vector field; automorphism group; real hypersurface; infinite type point AMS Subject Classifications: Primary 32M05; Secondary 32H02; 32H50; 32T25 Introduction Let M be a C ∞ -smooth real hypersurface in Cn and p ∈ M We denote by Aut(M, p) the stability group of M, that is, those germs at p of biholomorphisms mapping M into itself and fixing p We also denote by hol0 (M, p) the set of all germs at p of holomorphic vector fields in Cn vanishing at p whose real part is tangent to M For real hypersurfaces in Cn , the study of stability groups of various hypersurfaces with further assumptions is given in [1–9] Recently, Koláˇr and Meylan [3] and Koláˇr, Meylan and Zaitsev [7] obtained a precise description of the derivatives needed to characterize an automorphism of a general hypersurface However, these results are known for Levi nondegenerate hypersurfaces or more generally for Levi degenerate hypersurfaces of D’Angelo finite type (cf [10]) Throughout the article, we consider C ∞ -smooth real hypersurfaces of D’Angelo infinite type in C2 We shall describe the stability groups of M(a, α, p, q) (defined below) and a radially symmetric hypersurface in C2 , which are showed in [11,12] that they admit non-zero tangential holomorphic vector fields vanishing at infinite type points n ∗ Given a non-zero holomorphic function a(z) = ∞ n=1 an z (an ∈ C, ∀ n ∈ N ) defined ∞ on := {z ∈ C : |z| < } ( > 0), C -smooth functions p, q defined respectively on (0, ) and [0, ) satisfying that q(0) = and that the function ∗ Email: thunv@vnu.edu.vn Current address: Center for Geometry and its Applications, Pohang University of Science and Technology, Pohang 790–784, The Republic of Korea © 2014 Taylor & Francis N Van Thu g(z) = if < |z| < if z = e p(|z|) 0 is C ∞ -smooth and vanishes to infinite order at z = 0, and an α ∈ R, we denote by M(a, α, p, q) the germ at (0, 0) of a real hypersurface defined by ρ(z , z ) := Re z + P(z ) + F(z , Im z ) = 0, Downloaded by [University of Otago] at 19:36 26 December 2014 where F and P are respectively defined on × (−δ0 , δ0 ) ( , δ0 > small enough) and by ⎧ cos R(z )+αt ⎨ if α = − α log cos(R(z )) F(z , t) = ⎩ tan(R(z ))t if α = 0, where R(z ) = q(|z |) − Re ∞ an n n=1 n z P(z ) = for all z ∈ log [1 + α P1 (z )] P1 (z ) α where ∞ P1 (z ) = exp p(|z |) + Re n=1 , and if α = if α = 0, an n z − log cos R(z ) in for all z ∈ ∗0 and P1 (0) = Then we can see that P, F are C ∞ -smooth respectively in and × (−δ0 , δ0 ) and P vanishes to infinite order at 0, and hence M(a, α, p, q) is C ∞ -smooth and of infinite type in the sense of D’Angelo (cf [10]) In [12], the author proved the following theorem Theorem [12] hol0 M(a, α, p, q), is generated by H a,α (z , z ) = L α (z )a(z ) where L α (z ) = α z1 ∂ ∂ + i z2 , ∂z ∂z exp(αz ) − if α = if α = It is also shown in [12] that if M is a C ∞ -smooth hypersurface in C2 satisfying that P is positive on a punctured disk, P vanishes to infinite order at 0, and F(z , t) is real-analytic in a neighbourhood of (0, 0) in C × R, then hol0 (M, 0) = if and only if, after a change of variable in z , M = M(a, α, p, q) for some a, α, p, q We let φta,α (t ∈ R) denote the holomorphic map defined on a neighbourhood U of the origin in C2 by setting ⎧ ⎨ − log + (e−αz − 1) exp t a(z eiτ )dτ , z eit if α = 0 α a,α φt (z , z ) = t iτ it ⎩ z exp if α = 0 a(z e )dτ , z e By shrinking U if necessary we can see that φta,α (t ∈ R) is well-defined In addition, each φta,α preserves M(a, α, p, q) (see cf Theorem in Appendix) Moreover, it is easily Complex Variables and Elliptic Equations checked that {φta,α }t∈R is a one-parameter subgroup of Aut M(a, α, p, q), , which is generated by the holomorphic vector field H a,α The first aim of this paper is to prove the following theorem, which gives an explicit description for the stability group of the hypersurface M(a, α, p, q) Theorem A Aut M(a, α, p, q), = {φta,α | t ∈ R} For the case that M is radially symmetric, Byun et al [11] obtained the following theorem Downloaded by [University of Otago] at 19:36 26 December 2014 Theorem [11] Let (M, 0) be a real C ∞ -smooth hypersurface germ at defined by the equation ρ(z) := ρ(z , z ) = Re z + P(z ) + Im z Q(z , Im z ) = satisfying the conditions: (i) (ii) (iii) (iv) P, Q are C ∞ -smooth with P(0) = Q(0, 0) = 0, P(z ) = P(|z |), Q(z , t) = Q(|z |, t) for any z and t, P(z ) > for any z = 0, and P(z ) vanishes to infinite order at z = Then hol0 (M, 0) = iβz ∂z∂ : β ∈ R We note that the condition (iv) simply says that is a point of D’Angelo infinite type Now let us denote by {Rt }t∈R the one-parameter subgroup of Aut(M, 0) generated by the holomorphic vector field H R (z , z ) = i z ∂z∂ , that is, Rt (z , z ) = z , z eit , ∀t ∈ R The second aim of this paper is to show that the stability group of a radially symmetric hypersurface of infinite type in C2 is exactly the one-parameter group {Rt }t∈R Namely, we prove the following theorem Theorem B Let (M, 0) be a real C ∞ -smooth hypersurface germ at defined by the equation ρ(z) := ρ(z , z ) = Re z + P(z ) + Im z Q(z , Im z ) = satisfying the conditions: (i) (ii) (iii) (iv) P, Q are C ∞ -smooth with P(0) = Q(0, 0) = 0, P(z ) = P(|z |), Q(z , t) = Q(|z |, t) for any z and t, P(z ) > for any z = 0, and P(z ) vanishes to infinite order at z = Then Aut(M, 0) = {Rt | t ∈ R} This paper is organized as follows In Section 2, we give several properties of functions vanishing to infinite order at the origin In Section 3, we prove Theorem A Section is devoted to the proof of Theorem B Finally, a theorem is pointed out in Appendix 4 N Van Thu Preliminaries In this section, we will recall the definition of function vanishing to infinite order at the origin in the complex plane and we will introduce several lemmas used to prove Theorems A and B Definition We say that a C ∞ -smooth function P : U (0) → R on a neighbourhood U (0) of the origin in Rn vanishes to infinite order at if ∂ α1 +···+αn P(0) = ∂ x1α1 · · · ∂ xnαn Downloaded by [University of Otago] at 19:36 26 December 2014 for every index α = (α1 , αn ) ∈ Nn Lem m a Let P : U (0) → R be a C ∞ -smooth function on a neighbourhood U (0) of the origin in Rn Then P vanishes to infinite order at if and only if lim (x1 , ,xn )→(0, ,0) P(x1 , , xn ) =0 |x1 |α1 · · · |xn |αn for any index α = (α1 , , αn ) ∈ Nn Proof The proof follows easily from Taylor’s theorem Corollary If a C ∞ -smooth function P on a neighbourhood of the origin in Rn α +···+αn vanishes to infinite order at 0, then ∂ α11 αn P(x , , x n ) does also for any index α = (α1 , αn ) ∈ Nn Lem m a for all z ∈ ∂ x1 ···∂ xn Suppose that P ∈ C ∞ ( ) ( > 0) vanishes to infinite order at 0, P(z) > ∗ := {z ∈ : z = 0}, and there are α > and β > such that lim z→0 P(αz) = β P(z) Then α = β = Proof have Suppose that there exist α > and β > such that lim z→0 P(αz) P(z) = β Then, we P(αz) = β + γ (z), P(z) where γ is a function defined on with γ (z) → as z → Since γ (z) → as z → 0, there exists δ0 > such that |γ (z)| < β/2 for any z ∈ δ0 We consider the following cases Complex Variables and Elliptic Equations ∗ δ0 Case < α < In this case, fix z ∈ Then for each positive integer n, we get P(α n z ) P(α n z ) P(αz ) = ··· n−1 P(z ) P(z ) P(α z ) · · · (β + γ (z )) = β + γ α n−1 z · · · (β − |γ (z )|) ≥ β − γ α n−1 z ≥ (β/2) n (1) Downloaded by [University of Otago] at 19:36 26 December 2014 Moreover, let us choose a positive integer m such that α m < β/2 Then it follows from (1) that P(α n z ) P(z ) β/2 n ≥ (2) m |z |m α m α n |z | z0 ) This yields that (αP(α n |z |)m → +∞ as n → ∞, which contradicts the fact that P vanishes to infinite order at P( α1 z) Case < α Since lim z→0 P(αz) P(z) = β, it follows that lim z→0 P(z) = β Following case 1, it is impossible Therefore, α = 1, and thus it is obvious that β = The proof is complete n Lem m a Let p(t) be a C ∞ -smooth function on (0, P(z) = 0) ( > 0) such that the function if z ∈ ∗0 if z = e p(|z|) vanishes to infinite order at z = Let β ∈ C ∞ ( ) with β(0) = Then P(|z + zβ(z)|) − P(|z|) = P(|z|) |z| p (|z|) (Re(β(z) + o(β(z))) + o((β(z))2 ) for any z ∈ Proof ∗ satisfying z + zβ(z) ∈ By Taylor’s theorem, for any z ∈ ∗ satisfying z + zβ(z) ∈ we have P (ξz ) P (|z|) |z + zβ(z)| − |z| + |z + zβ(z)| − |z| 1! for some real number ξz between |z| and |z + zβ(z)| On the other hand, P(|z + zβ(z)|) = P(|z|) + 2|z|2 Re(β(z)) + |z|2 |β(z)|2 |z + zβ(z)|2 − |z|2 = |z + zβ(z)| + |z| |z + zβ(z)| + |z| = |z| Re(β(z)) + o(β(z)) (3) |z + zβ(z)| − |z| = Moreover, P (|z|) = P(|z|) p (|z|) for all z ∈ the proof follows from (3) to (4) ∗ (4) and P (ξz ) → as z → Therefore, Lem m a Let P(z) = e p(|z|)+g(z) be a C ∞ -smooth function on ( > 0) vanishing to infinite order at z = 0, where g ∈ C ∞ ( ) and p ∈ C ∞ (0, ) Let β ∈ C ∞ ( ) with β(z) = O(P(z)) Then P(z + zβ(z)) − P(z) = P(z) |z| p (|z|) Re(β(z)) + o(β(z)) + o(β(z)) N Van Thu for any z ∈ Proof ∗ satisfying z + zβ(z) ∈ Since β(z) = O(P(z)), by Lemma one has e p(|z+zβ(z)|) = + |z| p (|z|) Re(β(z)) + o(β(z)) + o(β(z)) e p(|z|) for any z ∈ ∗ satisfying z + zβ(z) ∈ P(z + zβ(z)) − P(z) = P(z) Downloaded by [University of Otago] at 19:36 26 December 2014 = P(z) Then we obtain e p(|z+zβ(z)|) g(z+zβ(z))−g(z) e −1 e p(|z|) + |z| p (|z|) Re(β(z)) + o(β(z)) + o(β(z)) e g(z+zβ(z))−g(z) − = P(z) |z| p (|z|) Re(β(z)) + o(β(z)) + o(β(z)) for any z ∈ ∗ satisfying z + zβ(z) ∈ This ends the proof Stability group of M(a, α, p, q) This section is entirely devoted to the proof of Theorem A Let a, α, , δ0 , F, P, P1 , p, q be given as in Section In what follows, F can be written as F(z , t) = t Q(z , t), where Q is C ∞ -smooth satisfying Q(0, 0) = For a proof of Theorem A, we need the following lemmas Lem m a If f = ( f , f ) ∈ Aut M(a, α, p, q), satisfying f (z , z ) = αz + ∞ k j ∗ k, j=1 bk j z z , where α > and bk j ∈ C (k, j ∈ N ), then α = and f (z , z ) = z + o(z ) Proof Expand f into Taylor series, one gets ∞ f (z , z ) = j ak j z 1k z , k, j=0 where a jk ∈ C ( j, k ∈ N) Note that a00 = f (0, 0) = Since f (M(a, α, p, q)) ⊂ M(a, α, p, q), we have ⎞ ⎛ Re ⎝ ∞ ak j it − P(z ) − t Q(z , t) z ⎠ k j k, j=0 ⎛ + P ⎝αz + ⎛ + Im ⎝ ⎛ ⎞ ∞ bk j it − P(z ) − t Q(z , t) k, j=1 ∞ k j⎠ z2 ⎞ ak j it − P(z ) − t Q(z , t) z ⎠ k j k, j=0 × Q ⎝αz + ∞ j bk j (it − P(z ) − t Q(z , t))k z , k, j=1 Complex Variables and Elliptic Equations ⎛ ⎞⎞ Im ⎝ ∞ ak j (it − P(z ) − t Q(z , t))k z ⎠⎠ ≡ j (5) k, j=0 × (−δ0 , δ0 ) for some , δ0 > We now consider the following cases Case f (0, z ) ≡ In this case, there is j1 ∈ N∗ such that a0 j1 = and f (z , z ) = j j a0 j1 z 21 + o(z 21 ) + O(z ) Since P(z ) = o(|z | j1 ), letting t = in (5), one deduces that j1 Re(a0 j1 z ) + o(|z | j1 ) ≡ on , which is impossible Case f (0, z ) ≡ We can write f (z , z ) = βz + o(z ), where β ∈ C∗ By (5), we get on Downloaded by [University of Otago] at 19:36 26 December 2014 Re β(it − P(z ) − t Q(z , t)) + o it − P(z ) − t Q(z , t) + P αz + z O(it − P(z ) − t Q(z , t)) + Im (β(it − P(z ) − t Q(z , t)) + o(it − P(z ) − t Q(z , t))) × Q (αz + z O(it − P(z ) − t Q(z , t)), Im β(it − P(z ) − t Q(z , t)) + o(it − P(z ) − t Q(z , t)) ≡0 (6) on × (−δ0 , δ0 ) In particular, inserting z = into (6) one has Re(βi) + O(t) ≡ 0, and this thus implies Im(β) = On the other hand, letting t = in (6) we obtain −Re(β)P(z ) + P αz + z O(P(z )) + o(P(z )) ≡ on This yields that lim z →0 P αz + z O(P(z )) /P(z ) = Re(β) > By Lemma and the fact that P(z ) p (|z |) vanishes to infinite order at z = (cf Corollary 1), we deduce that lim z →0 P αz + z O(P(z )) P(αz ) = lim = Re(β) > z →0 P(z ) P(z ) Therefore, by Lemma 2, we conclude that α = β = The proof is now complete ∞ k Lem m a If f ∈ Aut M(a, α, p, q), satisfying f (z , z ) = z + ∞ k=1 j=0 ak j z z k j with a10 = and f (z , z ) = z + ∞ k, j=1 bk j z z , where ak j , bk j ∈ C (k, j ∈ N), then f = id j Proof Since f preserves M(a, α, p, q), it follows that ⎛ Re ⎝(it − P(z ) − t Q(z , t)) + ⎛ + P ⎝z + ⎛ ∞ ∞ ak j (it − k=1 j=0 ∞ ⎞ j P(z ) − t Q(z , t))k z ⎠ ⎞ bk j (it − P(z ) − t Q(z , t))k z ⎠ j k, j=1 + Im ⎝(it − P(z ) − t Q(z , t)) + ∞ ∞ k=1 j=0 ⎞ ak j (it − P(z ) − t Q(z , t))k z ⎠ j N Van Thu ⎛ ∞ × Q ⎝z + j bk j (it − P(z ) − t Q(z , t))k z , k, j=1 ⎛ Im ⎝(it − P(z ) − t Q(z , t)) + ∞ ⎞⎞ ∞ ak j (it − j P(z ) − t Q(z , t))k z ⎠⎠ ≡ 0, k=1 j=0 (7) or equivalently, ⎛ Re ⎝ ∞ ∞ ⎞ ak j (it − P(z ) − t Q(z , t))k z ⎠ j Downloaded by [University of Otago] at 19:36 26 December 2014 k=1 j=0 ⎛ + P ⎝z + ⎡ ⎛ ⎞ ∞ bk j (it − ∞ k, j=1 ∞ t + Im ⎝ + Im ⎝ ⎛ j bk j (it − P(z ) − t Q(z , t))k z , ⎛ ∞ ⎞⎞ ⎤ j ak j (it − P(z ) − t Q(z , t))k z ⎠⎠ − Q(z , t)⎦ k=1 j=0 ∞ P(z ) k, j=1 + t ⎣ Q ⎝z + ⎛ j P(z ) − t Q(z , t))k z ⎠ − ⎞ ∞ ak j (it − j P(z ) − t Q(z , t))k z ⎠ k=1 j=0 × Q ⎝z + ∞ j bk j (it − P(z ) − t Q(z , t))k z , k, j=1 ⎛ t + Im ⎝ ∞ ∞ ⎞⎞ ak j (it − P(z ) − t Q(z , t))k z ⎠⎠ ≡ j (8) k=1 j=0 × (−δ0 , δ0 ) for some , δ0 > If f (z , z ) ≡ z , then let k1 = +∞ In the contrary case, let k1 be the smallest integer k such that ak j = for some j ∈ N∗ Then let j1 be the smallest integer j such that ak1 j = Similarly, if f (z , z ) ≡ z , then denote by k2 = +∞ Otherwise, let k2 be the smallest integer k such that bk j = for some j ∈ N∗ Denote by j2 the smallest integer j such that bk2 j = Since P(z ) = o(|z| j ) for any j ∈ N, inserting t = α P(z ) into (8) (with α ∈ R to be chosen later) one gets on j Re ak1 j1 P k1 (z )(αi − 1)k1 z 21 + o(|z | j1 ) j + P z + bk2 j2 P k2 (z )(αi − 1)k2 z 22 + o(|z | j2 ) j − P(z ) + α P(z ) Q z + bk2 j2 P k2 (z )(αi − 1)k2 z 22 + o(|z | j2 ) , Complex Variables and Elliptic Equations j α P(z ) + Im ak1 j1 P k1 (z )(αi − 1)k1 z 21 + o(|z | j1 ) − Q(z , α P(z )) j + Im ak1 j1 P k1 (z )(αi − 1)k1 z 21 + o(|z | j1 ) j × Q z + bk2 j2 P k2 (z )(αi − 1)k2 z 22 + o(|z | j2 ) , j α P(z ) + Im ak1 j1 P k1 (z )(αi − 1)k1 z 21 + o(|z | j1 ) on j j Downloaded by [University of Otago] at 19:36 26 December 2014 + Re ak1 j1 P k1 (z )(αi − 1)k1 z 21 + o(|z | j1 ) (9) Since Q(0, 0) = 0, (9) tells us that P z + bk2 j2 P k2 (z )(αi − 1)k2 z 22 + o(|z | j2 ) on ≡0 − P(z ) + P k2 +1 (z )o(|z | j2 ) ≡ (10) Moreover, one has by Lemma that j −1 P k2 +1 (z ) |z | p (|z |) Re bk2 j2 (αi − 1)k2 z 22 + o(|z | j2 −1 ) + o(|z | j2 −1 ) j + P k1 (z )Re ak1 j1 (αi − 1)k1 z 21 + o(|z | j1 ) ≡0 (11) We now observe that lim supr →0+ |r p (r )| = +∞, for otherwise one gets | p(r )| | log(r )| for every < r < , and thus P does not vanish to infinite order at We thus divide the proof into two cases as follows Case k2 < +∞ and k2 + < k1 ≤ +∞ In this case, P k1 (z ) = o |z | j2 P k2 +1 (z ) and hence (11) yields on j −1 P k2 +1 (z ) |z | p (|z |) Re bk2 j2 (αi − 1)k2 z 22 + o(|z | j2 −1 ) ≡ o |z | j2 P k2 +1 (z ) (12) on It is absurd Case k1 < +∞ and k1 − ≤ k2 ≤ +∞ By the fact that P(z ) p (|z |) vanishes to infinite order at z = (see Corollary 1), Lemma 4, and Equation (11), it follows that j Re ak1 j1 ((αi − 1)k1 z 21 + o(|z | j1 ) ≡ (13) on Notice that if j1 = 0, then k1 ≥ and α can thus be chosen so that Re(ak1 j1 (αi − 1)k1 ) = Therefore, Equation (13) is a contradiction Case k2 + = k1 ≤ +∞ Since k2 + = k1 < +∞, we have by (11) j −1 h(z ) := |z | p (|z |) Re bk2 j2 (αi − 1)k2 z 22 j + Re ak1 j1 (αi − 1)k1 z 21 + o(|z | j1 ) + o(|z | j2 −1 ) + o(|z | j2 −1 ) ≡ (14) on Because lim supr →0+ r p (r ) = +∞, j2 − = j1 + d for some d ∈ N∗ Theorefore taking lim r →0+ r 1j1 h(r eiθ ) for each θ ∈ R, from (9) one obtains Re c1 ei( j1 +d)θ = Re c2 ei j1 θ 10 N Van Thu for every θ ∈ R, where c1 , c2 ∈ C∗ This is impossible since {1, cos θ, sin θ, , cos(( j1 + d)θ ), sin(( j1 + d)θ )} are linearly independent Altogether, we conclude that k1 = k2 = +∞, and hence the proof is complete Now we are ready to prove Theorem A Proof of Theorem A For f = ( f , f ) ∈ Aut M(a, α, p, q), , we let {Ft }t∈R be the a,α family of automorphisms by setting Ft := f ◦ φ−t ◦ f −1 Then it follows that {Ft }t∈R is a one-parameter subgroup of Aut M(a, α, p, q), By Theorem 1, there exists a real a,α for all t ∈ R This implies that number δ such that Ft = φδt Downloaded by [University of Otago] at 19:36 26 December 2014 a,α f = φδt ◦ f ◦ φta,α , ∀t ∈ R (15) We note that if δ = 0, then f = f ◦ φta,α and thus φta,α = id for any t ∈ R, which is a contradiction Hence, we may assume that δ = Now we shall prove that δ = −1 Indeed, we have by (15) f (z , z ) ≡ eiδt f − eiδt α log + (e−αz − 1) exp f z exp( t a(z eiτ )dτ ), z t a(z eiτ )dτ , z eit if α = if α = eit (16) on a neighbourhood U of (0, 0) ∈ C2 and for all t ∈ R Expand f into Taylor series, one obtains that ∞ f (z , z ) = j bk j z 1k z , k, j=0 where bk j ∈ C (k, j ∈ N) and b00 = f (0, 0) = We claim that bk0 = for every k ∈ N∗ Suppose, to derive a contradiction, that there exists the smallest integer k0 ∈ N∗ such that bk0 = Then putting z = in (16) and noting that a(0) = 0, we obtain bk0 z 1k0 + o z 1k0 ≡ eiδt bk0 z 1k0 + o z 1k0 on the set {z ∈ C | (z , 0) ∈ U } for all t ∈ R This is absurd since δ = Therefore, bk0 = for every k = 1, 2, Besides, since f is a biholomorphism, we get b01 = On the other hand, letting z = in (16) one gets ∞ ∞ j b0 j z ≡ j b0 j z ei( j+δ)t j=1 j=1 on {z ∈ C | (0, z ) ∈ U } for all t ∈ R Taking the derivative both sides of the above equation with respect to t at t = 0, we arrive at ∞ j b0 j z i j + δ ≡ (17) j=1 on {z ∈ C | (0, z ) ∈ U } Since b01 = 0, (17) entails that δ = −1 and furthermore b0 j = for all j = 2, 3, In addition, replacing f by f ◦ φθa,α for a reasonable θ ∈ R, we can k j assume that b01 = α > 0, and thus f (z ) = αz + ∞ k, j=1 bk j z z Complex Variables and Elliptic Equations 11 Applying Lemma 5, we conclude that f (z , z ) = z + o(z ) and f (z , z ) = z + O(z z ) Finally, Lemma ensures that f = id, and thus the proof is complete Stability groups of radially symmetric hypersurfaces of infinite type In this section, we are going to prove Theorem B To this, let M be a C ∞ -smooth hypersurface as in Theorem B That is, M is defined by Downloaded by [University of Otago] at 19:36 26 December 2014 ρ(z , z ) = Re z + P(z ) + Im z Q(z , Im z ) = 0, where P, Q are C ∞ -smooth functions on and ×(−δ0 , δ0 ) ( , δ0 > 0), respectively, satisfying conditions (i)–(iv) as in Theorem B In order to prove Theorem B, we need the following lemma ∞ k Lem m a If f ∈ Aut(M, 0) satisfying f (z , z ) = k=1 ak z and f (z , z ) = j ∞ z j=0 b j z , where ak , b j ∈ C (k, j ∈ N), b0 > and a1 = 0, then a1 = b0 = Proof Since M is invariant under f , we have ∞ ak it − P(z ) − t Q(z , t) Re k k=1 ⎛ + P ⎝z ⎞ ∞ b j it − P(z ) − t Q(z , t) j⎠ j=0 ∞ + Im ak (it − P(z ) − t Q(z , t))k ⎛ k=1 × Q ⎝z ∞ b j (it − P(z ) − t Q(z , t)) j , j=0 ∞ ak (it − P(z ) − t Q(z , t))k Im ≡0 (18) k=1 on × (−δ0 , δ0 ) for some , δ0 > small enough It follows from (18) with z = that Re(a1 it) + o(t) = for every t ∈ R small enough This yields that Im(a1 ) = On the other hand, inserting t = into (18) one has P (b0 z + z O(P(z ))) − Re(a1 )P(z ) + o(P(z )) ≡ (19) This implies that lim z →0 P b0 z + z O(P(z )) /P(z ) = Re(a1 ) = a1 > By assumption, we can write P(z ) = e p(|z |) for all z ∈ ∗0 for some function p ∈ C ∞ (0, ) with limt→0+ p(t) = −∞ such that P vanishes to infinite order at z = Therefore, by Lemma and the fact that P(z ) p (|z |) vanishes to infinite order at z = on 12 N Van Thu (cf Corollary 1), one gets that lim z →0 P b0 z + z O(P(z )) P(b0 z ) = lim = a1 > z →0 P(z ) P(z ) Hence, Lemma ensures that a1 = b0 = 1, which ends the proof Proof of Theorem B For f = ( f , f ) ∈ Aut(M, 0) We define Ft by setting Ft := f ◦ R−t ◦ f −1 for each t ∈ R Then {Ft }t∈R is a one-parameter subgroup of Aut(M, 0) Using the same arguments as in the proof of Theorem A, Theorem yields that Ft = R−t for all t ∈ R This implies that Downloaded by [University of Otago] at 19:36 26 December 2014 f = R−t ◦ f ◦ Rt , ∀t ∈ R, (20) namely f (z , z ) ≡ f (z , z eit ) f (z , z ) ≡ e−it f (z , z eit ) on a neighbourhood U of (0, 0) in C2 for all t ∈ R Indeed, this tells us that f (z , z ) = j ∞ ∞ k k=1 ak z and f (z , z ) = z j=0 b j z for all (z , z ) ∈ U , where ak , b j ∈ C for all ∗ j ∈ N and k ∈ N We note that a1 , b0 ∈ C∗ In addition, replacing f by f ◦ Rθ for some θ ∈ R, we can assume that b0 is a positive real number We now apply Lemma to obtain that a1 = b0 = Finally, by Lemma we conclude that f = id (Lemma still holds for a C ∞ -smooth radially symmetric hypersurface satisfying (i)–(iv).) Hence, the proof is complete Acknowledgements The author would like to thank Prof Do Duc Thai for his precious discussions on this material Funding The research of the author was supported in part by an NRF [grant number 2011-0030044] (SRCGAIA) of the Ministry of Education, The Republic of Korea References [1] Chern SS, Moser JK Real hypersurfaces in complex manifolds Acta Math 1974;133:219–271 [2] Ezhov V, Koláˇr M, Schmalz G Degenerate hypersurfaces with a two-parametric family of automorphisms Complex Var Elliptic Equ 2009;54:283–291 [3] Koláˇr M, Meylan F Infinitesimal CR automorphisms of hypersurfaces of finite type in C2 Arch Math (Brno) 2011;47:367–375 [4] Koláˇr M Local equivalence of symmetric hypersurfaces in C2 Trans Am Math Soc 2010;362:2833–2843 [5] Koláˇr M Local symmetries of finite type hypersurfaces in C2 Sci China Ser A 2006;49:1633– 1641 [6] Koláˇr M Normal forms for hypersurfaces of finite type in C2 Math Res Lett 2005;12:897–910 [7] Koláˇr M, Meylan F, Zaitsev D Chern–Moser operators and polynomial models in CR geometry Adv Math 2014;263:321–356 Complex Variables and Elliptic Equations 13 [8] Stanton N Infinitesimal CR automorphisms of real hypersurfaces Am J Math 1996;118:209– 233 [9] Stanton N Infinitesimal CR automorphisms of rigid hypersurfaces Am J Math 1995;117:141– 167 [10] D’Angelo JP Real hypersurfaces, orders of contact, and applications Ann Math 1982;115:615– 637 [11] Byun J, Joo J-C, Song M The characterization of holomorphic vector fields vanishing at an infinite type point J Math Anal Appl 2012;387:667–675 [12] Ninh VT On the existence of tangential holomorphic vector fields vanishing at an infinite type point arXiv:1303.6156v7 Downloaded by [University of Otago] at 19:36 26 December 2014 Appendix Theorem Let p0 ∈ M(a, α, p, q) be arbitrary Then any flow of the holomorphic vector field ∂ ∂ + i z2 , H a,α (z , z ) = L α (z )a(z ) ∂z ∂z where L α (z ) = α exp(αz ) − z1 if α = if α = 0, starting from po is contained in M(a, α, p, q) Proof Let P1 , P, R, F be functions and > 0, δ0 > be positive real numbers introduced to define M(a, α, p, q) and let Q (z ) := tan(R(z )) for all z ∈ Then by Lemmas and 8, and Corollary in [12, Appendix A] we have the following equations 1 + Q 20 (z ) ia(z ) ≡ 0; Q (z ) + a(z )P1 (z ) ≡ 0; (ii) Re i z P1z (z ) − 2i (i) Re i z Q 0z (z ) + (iii) Re i z Pz (z ) + exp − α P(z ) − α (iv) (i + Ft (z , t) exp α it − F(z , t) Q (z ) + a(z ) ≡ for α = 0; 2i ≡ i + Q (z ); (v) Re 2iαz Fz (z , t) + (Ft (z , t) − Q (z )) ia(z ) ≡ 0 for any t ∈ (−δ0 , δ0 ) Let z(t) = (z (t), z (t)), −∞ < t < +∞, be the flow of H a,α satisfying z(0) = p0 This means that on z (t) = L(z (t))a(z (t)) z (t) = i z (t) for all t ∈ R Let g(t) := ρ(z (t), z (t)), −∞ < t < +∞ Then g (t) = 2Re ρz (z(t))z (t) + ρz (z(t))z (t) , We divide the proof into two cases ∀t ∈ R 14 N Van Thu (a) α = In this case, F(z , τ ) = Q (z )τ for all (z , τ ) ∈ (i) and (ii) one obtains that g (t) = 2Re × (−δ0 , δ0 ) Therefore, by Q (z (t)) + z (t)a(z (t)) 2i + P1z (z (t)) + (Im z (t))Q 0z (z (t)) iβz (t) = 2Re Q (z (t)) + (i(Im z (t)) + g(t) − P1 (z (t)) 2i −(Im z )Q (z (t))) a(z (t)) + P1z (z (t)) + (Im z )Q 0z (z (t)) i z (t) Q (z (t)) + a(z (t))P1 (z (t)) 2i 1 + Q (z (t))2 ia(z (t)) + (Im z (t))Re i z (t)Q 0z (z (t)) + Q (z (t)) + a(z (t)) + 2g(t)Re 2i Q (z (t)) + a(z (t)) = 2g(t)Re 2i Downloaded by [University of Otago] at 19:36 26 December 2014 = 2Re i z (t)P1z (z (t)) − for every t ∈ R Since g(0) = ρ( p0 ) = 0, by the uniqueness of the solution of differential equations, we conclude that g(t) ≡ This proves the theorem for α = (b) α = It follows from (iii), to (v) that g (t) = 2Re Fτ (z (t), Im z (t)) + 2i L(z (t))a(z (t)) + Pz (z (t)) + Fz (z (t), Im z (t)) i z (t) = 2Re Fτ (z (t), Im z (t)) + 2i −F(z (t), Im z (t)) exp α iIm z (t) + g(t) − P(z (t)) α − a(z (t)) + Pz (z (t)) + Fz (z (t), Im z (t)) i z (t) i + Fτ (z (t), Im z (t)) exp α iIm z (t) − F(z (t), Im z (t)) α 2i Fτ (z (t), Im z (t)) 1 + a(z (t)) × exp(−α P(z (t))) exp(αg(t))a(z (t)) − α 2i = 2Re + Pz (z (t)) + Fz (z (t), Im z (t)) i z (t) i + Q (z (t)) exp(−α P(z (t))) exp(αg(t))a(z (t)) α 2i Fτ (z (t), Im z (t)) 1 + a(z (t)) + Pz (z (t)) + Fz (z (t), Im z (t)) i z (t) − α 2i Q (z (t)) exp(−α P(z (t))) − 1 + a(z (t)) = 2Re i z (t)Pz (z (t)) + 2i α + 2Re i z (t)Fz (z (t), Im z (t)) + (Fτ (z (t), Im z (t)) − Q (z (t))) ia(z (t)) 2α = 2Re Complex Variables and Elliptic Equations 15 Q (z (t)) exp(αg(t)) − 1 exp(−α P(z (t)))Re + a(z (t)) α 2i Q (z (t)) exp(αg(t)) − a(z (t)) exp(−α P(z (t)))Re + =2 α 2i +2 Downloaded by [University of Otago] at 19:36 26 December 2014 for every t ∈ R Since g(0) = ρ( p0 ) = 0, again by the uniqueness of the solution of differential equations, we conclude that g(t) ≡ This ends the proof  ... Equations, 20 14 http://dx.doi.org/10.1080/17476933 .20 14.986656 On the CR automorphism group of a certain hypersurface of infinite type in C2 Ninh Van Thu∗1 Department of Mathematics, Vietnam National... 20 14 ;26 3: 321 –356 Complex Variables and Elliptic Equations 13 [8] Stanton N Infinitesimal CR automorphisms of real hypersurfaces Am J Math 1996;118 :20 9– 23 3 [9] Stanton N Infinitesimal CR automorphisms... two-parametric family of automorphisms Complex Var Elliptic Equ 20 09;54 :28 3 29 1 [3] Koláˇr M, Meylan F Infinitesimal CR automorphisms of hypersurfaces of finite type in C2 Arch Math (Brno) 20 11;47:367–375