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

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On the CR automorphism group of a certain hypersurface of infinite type in

ℂ 2

Ninh Van Thua a

Department of Mathematics, Vietnam National University, 334 Nguyen Trai Str., Hanoi, Vietnam

Published online: 24 Dec 2014

To cite this article: Ninh Van Thu (2014): On the CR automorphism group of a certain hypersurface

of infinite type in ℂ2

, 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

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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

(Received 1 September 2014; accepted 7 November 2014)

We considerC∞-smooth real hypersurfaces of infinite type inC2 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

1 Introduction

Let M be a C∞-smooth real hypersurface inCn 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 inCn vanishing at p whose real part is tangent to M.

For real hypersurfaces inCn, 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 non-degenerate hypersurfaces or more generally for Levi non-degenerate hypersurfaces of D’Angelo finite type (cf [10])

Throughout the article, we considerC∞-smooth real hypersurfaces of D’Angelo infinite type inC2 We shall describe the stability groups of M (a, α, p, q) (defined below) and a

radially symmetric hypersurface inC2, which are showed in [11,12] that they admit non-zero tangential holomorphic vector fields vanishing at infinite type points

Given a non-zero holomorphic function a (z) =∞n=1a n z n (a n ∈ C, ∀ n ∈ N∗) defined

on 0 := {z ∈ C: |z| < 0} (0> 0), C∞-smooth functions p , q defined respectively on (0, 0) and [0, 0) satisfying that q(0) = 0 and that the function

∗Email: thunv@vnu.edu.vn

1Current address: Center for Geometry and its Applications, Pohang University of Science and Technology, Pohang 790–784, The Republic of Korea

g (z) =



e p (|z|) if 0< |z| < 0

0 if 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(z1, z2) := Re z1+ P(z2) + F(z2, Im z1) = 0,

where F and P are respectively defined on  0× (−δ0, δ0) (0, δ0> 0 small enough) and

 0by

F (z2, t) =

⎧

⎨

⎩−

1

αlog



cos

R (z2)+αt cos(R(z2))



 if α = 0

where R (z2) = q(|z2|) − Re ∞

n=1a n n z n2

for all z2∈  0, and

P (z2) =

1

αlog [1+ αP1(z2)] ifα = 0

where

P1(z2) = exp p (|z2|) + Re

∞

n=1

a n

i n z

n

2 − logcos

R (z2) 

for all z2∈ ∗

0 and P1(0) = 0.

Then we can see that P , F are C∞-smooth respectively in 0and 0× (−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

Th e o r e m 1 [12] hol0



M (a, α, p, q), 0 is generated by

H a ,α (z1, z2) = L α (z1)a(z2) ∂

∂z1

+ iz2 ∂

∂z2

, where

L α (z1) =

1

α

 exp(αz1) − 1 if α = 0

It is also shown in [12] that if M is a C∞-smooth hypersurface inC2satisfying that P is positive on a punctured disk, P vanishes to infinite order at 0, and F (z2, t) is real-analytic

in a neighbourhood of(0, 0) in C × R, then hol0(M, 0) = 0 if and only if, after a change

of variable in z2, M = M(a, α, p, q) for some a, α, p, q.

We letφ a ,α

t (t ∈ R) denote the holomorphic map defined on a neighbourhood U of the

origin inC2by setting

φ a ,α

t (z1, z2) =

⎧

⎨

⎩



−1

αlog



1+ (e −αz1− 1) exp t

0a (z2e i τ )dτ , z2e i t

ifα = 0



z1exp t

0a (z2e i τ )dτ , z2e i t



ifα = 0.

By shrinking U if necessary we can see that φ a ,α

t (t ∈ R) is well-defined In addition,

eachφ a ,α preserves M (a, α, p, q) (see cf Theorem3in Appendix) Moreover, it is easily

Complex Variables and Elliptic Equations 3 checked that{φ a ,α

t }t∈R is a one-parameter subgroup of Aut

M (a, α, p, q), 0 , 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).

Th e o r e m A Aut

M (a, α, p, q), 0 = {φ a ,α

t | t ∈ R}.

For the case that M is radially symmetric, Byun et al [11] obtained the following theorem

Th e o r e m 2 [11] Let (M, 0) be a real C∞-smooth hypersurface germ at 0 defined by the

equation ρ(z) := ρ(z1, z2) = Re z1+ P(z2) + Im z1Q (z2, Im z1) = 0 satisfying the conditions:

(i) P , Q are C∞-smooth with P (0) = Q(0, 0) = 0,

(ii) P (z2) = P(|z2|), Q(z2, t) = Q(|z2|, t) for any z2and t,

(iii) P (z2) > 0 for any z2= 0, and

(iv) P (z2) vanishes to infinite order at z2= 0.

Then hol0(M, 0) =i βz2 ∂

∂z2: β ∈ R.

We note that the condition(iv) simply says that 0 is a point of D’Angelo infinite type.

Now let us denote by{R t}t∈Rthe one-parameter subgroup of Aut(M, 0) generated by the

holomorphic vector field H R (z1, z2) = iz2 ∂

∂z2, that is,

R t (z1, z2) =z1, z2e i t

, ∀t ∈ R.

The second aim of this paper is to show that the stability group of a radially symmetric hypersurface of infinite type inC2is exactly the one-parameter group{R t}t∈R Namely, we prove the following theorem

Th e o r e m B Let (M, 0) be a real C∞-smooth hypersurface germ at 0 defined by the

equation ρ(z) := ρ(z1, z2) = Re z1+ P(z2) + Im z1Q (z2, Im z1) = 0 satisfying the conditions:

(i) P , Q are C∞-smooth with P (0) = Q(0, 0) = 0,

(ii) P (z2) = P(|z2|), Q(z2, t) = Q(|z2|, t) for any z2and t,

(iii) P (z2) > 0 for any z2= 0, and

(iv) P (z2) vanishes to infinite order at z2= 0.

Then Aut(M, 0) = {R t | t ∈ R}.

This paper is organized as follows In Section2, we give several properties of functions vanishing to infinite order at the origin In Section3, we prove TheoremA Section4 is devoted to the proof of TheoremB Finally, a theorem is pointed out in Appendix

2 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

AandB

Definition 1 We say that aC∞-smooth function P : U(0) → R on a neighbourhood U(0)

of the origin inRnvanishes to infinite order at 0 if

∂ α1+···+α n

∂x α1

1 · · · ∂x α n

n

P (0) = 0

for every indexα = (α1, α n ) ∈ N n

Le m m a 1 Let P : U(0) → R be a C∞-smooth function on a neighbourhood U (0) of the origin inRn Then P vanishes to infinite order at 0 if and only if

lim

(x1, ,x n )→(0, ,0)

P (x1, , x n )

|x1|α1· · · |x n|α n = 0

for any index α = (α1, , α n ) ∈ N n

Co r o l l a r y 1 If a C∞-smooth function P on a neighbourhood of the origin in Rn

vanishes to infinite order at 0, then ∂ α1+···+αn

∂x1α1 ···∂x n αn

P (x1, , x n ) does also for any index α = (α1, α n ) ∈ N n

Le m m a 2 Suppose that P ∈ C∞( 0) (0> 0) vanishes to infinite order at 0, P(z) > 0 for all z ∈ ∗

0 := {z ∈  0: z = 0}, and there are α > 0 and β > 0 such that

lim

z→0

P (αz)

P (z) = β.

Then α = β = 1.

Proof Suppose that there existα > 0 and β > 0 such that lim z→0 P P (αz) (z) = β Then, we

have

P (αz)

P (z) = β + γ (z),

whereγ is a function defined on  0 withγ (z) → 0 as z → 0 Since γ (z) → 0 as z → 0,

there existsδ0> 0 such that |γ (z)| < β/2 for any z ∈  δ0

We consider the following cases

Complex Variables and Elliptic Equations 5

Case 1 0 < α < 1 In this case, fix z0∈ ∗

δ0 Then for each positive integer n, we get

P (α n z0)

P (z0) =

P (α n z0)

P (α n−1z0)· · ·

P (αz0)

P (z0)

=β + γα n−1z

0



· · · (β + γ (z0))

≥β −γ

α n−1z

0

· · · (β − |γ (z0)|)

Moreover, let us choose a positive integer m0such thatα m0 < β/2 Then it follows from

(1) that

P (α n z0)



α n |z0| m0 ≥ P (z0)

|z0|m0

β/2

α m0

n

This yields that P (α n z0)

(α n |z0|) m0 → +∞ as n → ∞, which contradicts the fact that P vanishes to

infinite order at 0

Case 2 1 < α Since lim z→0 P P (αz) (z) = β, it follows that lim z→0 P (

1

α z )

P (z) = 1

β Following case

1, it is impossible

Therefore,α = 1, and thus it is obvious that β = 1 The proof is complete. 

Le m m a 3 Let p (t) be a C∞-smooth function on (0, 0) (0> 0) such that the function

P (z) =



e p (|z|) if z ∈ ∗

0

0 if z= 0

vanishes to infinite order at z = 0 Let β ∈ C∞( 0) with β(0) = 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 ∈ ∗

0satisfying z + zβ(z) ∈  0 Proof By Taylor’s theorem, for any z ∈ ∗

0 satisfying z + zβ(z) ∈  0 we have

P (|z + zβ(z)|) = P(|z|) + P (|z|)

1!



|z + zβ(z)| − |z| + P (ξ z )

2



|z + zβ(z)| − |z| 2

(3) for some real numberξ zbetween|z| and |z + zβ(z)|.

On the other hand,

|z + zβ(z)| − |z| = |z + zβ(z)|2− |z|2

|z + zβ(z)| + |z| =

2|z|2Re(β(z)) + |z|2|β(z)|2

|z + zβ(z)| + |z|

Moreover, P (|z|) = P(|z|)p (|z|) for all z ∈ ∗

0 and P (ξ z ) → 0 as z → 0 Therefore,

Le m m a 4 Let P (z) = e p (|z|)+g(z) be a C∞-smooth function on  0 (0 > 0) vanishing

to infinite order at z = 0, where g ∈ C∞( 0) and p ∈ C∞(0, 0) Let β ∈ C∞( 0) 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))

for any z ∈ ∗

0satisfying z + zβ(z) ∈  0.

e p (|z+zβ(z)|)

e p (|z|) = 1 + |z|p (|z|)Re(β(z)) + o(β(z)) + o(β(z)) for any z ∈ ∗

0satisfying z + zβ(z) ∈  0 Then we obtain

P (z + zβ(z)) − P(z) = P(z) e p (|z+zβ(z)|)

e p (|z|) e g (z+zβ(z))−g(z)− 1



= P(z)

1+ |z|p (|z|)Re(β(z)) + o(β(z)) + o(β(z)) e g (z+zβ(z))−g(z)− 1

= P(z)|z|p (|z|)Re(β(z)) + o(β(z)) + o(β(z)) for any z ∈ ∗

0satisfying z + zβ(z) ∈  0 This ends the proof 

This section is entirely devoted to the proof of TheoremA Let a , α, 0, δ0, F, P, P1, p, q

be given as in Section1 In what follows, F can be written as F (z2, t) = t Q(z2, t), where

Q is C∞-smooth satisfying Q (0, 0) = 0 For a proof of TheoremA, we need the following lemmas

Le m m a 5 If f = ( f1, f2) ∈ AutM (a, α, p, q), 0 satisfying f2(z1, z2) = αz2 +

∞

k , j=1 b k j z k1z2j , where α > 0 and b k j ∈ C (k, j ∈ N∗), then α = 1 and f1(z1, z2) =

z1+ o(z1).

Proof Expand f1into Taylor series, one gets

f1(z1, z2) = ∞

k , j=0

a k j z k1z2j ,

where a j k ∈ C ( j, k ∈ N) Note that a00 = f1(0, 0) = 0 Since f (M(a, α, p, q)) ⊂

M (a, α, p, q), we have

Re

⎛

⎝∞

k , j=0

a k j



i t − P(z2) − t Q(z2, t) k

z2j

⎞

⎠

+ P

⎛

⎝αz2+ ∞

k , j=1

b k j

i t − P(z2) − t Q(z2, t) k

z2j

⎞

⎠ + Im

⎛

⎝∞

k , j=0

a k j

i t − P(z2) − t Q(z2, t) k

z2j

⎞

⎠

× Q

⎛

⎝αz2+

∞

k , j=1

b k j (it − P(z2) − t Q(z2, t)) k z2j ,

Complex Variables and Elliptic Equations 7

Im

⎛

⎝∞

k , j=0

a k j (it − P(z2) − t Q(z2, t)) k z2j

⎞

⎠

⎞

on 0× (−δ0, δ0) for some 0, δ0> 0.

We now consider the following cases

Case 1 f1(0, z2) ≡ 0 In this case, there is j1∈ N∗such that a 0 j

1 = 0 and f1(z1, z2) =

a 0 j1z j1

2 + o(z j1

2) + O(z1) Since P(z2) = o(|z2|j1), letting t = 0 in (5), one deduces that

Re(a 0 j1z j1

2) + o(|z2|j1) ≡ 0 on  0, which is impossible

Case 2 f1(0, z2) ≡ 0 We can write f1(z1, z1) = βz1+ o(z1), where β ∈ C∗ By (5), we get

Re

β(it − P(z2) − t Q(z2, t)) + oi t − P(z2) − t Q(z2, t)

+ Pαz2+ z2O (it − P(z2) − t Q(z2, t))

+ Im (β(it − P(z2) − t Q(z2, t)) + o(it − P(z2) − t Q(z2, t)))

× Q (αz2+ z2O (it − P(z2) − t Q(z2, t)),

Im

β(it − P(z2) − t Q(z2, t)) + o(it − P(z2) − t Q(z2, t)) ≡ 0 (6)

on 0× (−δ0, δ0) In particular, inserting z2= 0 into (6) one has Re(βi) + O(t) ≡ 0, and

this thus implies Im(β) = 0.

On the other hand, letting t= 0 in (6) we obtain

−Re(β)P(z2) + Pαz2+ z2O (P(z2)) + o(P(z2)) ≡ 0

on 0 This yields that limz2 →0P

αz2+ z2O (P(z2)) /P(z2) = Re(β) > 0 By Lemma4

and the fact that P (z2)p (|z2|) vanishes to infinite order at z2 = 0 (cf Corollary1), we deduce that

lim

z2→0

P (αz2)

P (z2) = limz2→0

P

αz2+ z2O (P(z2))

P (z2) = Re(β) > 0.

Therefore, by Lemma2, we conclude thatα = β = 1 The proof is now complete. 

Le m m a 6 If f ∈ AutM (a, α, p, q), 0 satisfying f1(z1, z2) = z1+∞k=1∞

j=0a k j z k1z2j with a10 = 0 and f2(z1, z2) = z2+∞k , j=1 b k j z k1z2j , where a k j , b k j ∈ C (k, j ∈ N), then

f = id.

Proof Since f preserves M (a, α, p, q), it follows that

Re

⎛

⎝(it − P(z2) − t Q(z2, t)) +∞

k=1

∞

j=0

a k j (it − P(z2) − t Q(z2, t)) k z2j

⎞

⎠

+ P

⎛

⎝z2+ ∞

k , j=1

b k j (it − P(z2) − t Q(z2, t)) k z2j

⎞

⎠ + Im

⎛

⎝(it − P(z2) − t Q(z2, t)) +

∞

k=1

∞

j=0

a k j (it − P(z2) − t Q(z2, t)) k z2j

⎞

⎠

× Q

⎛

⎝z2+ ∞

k , j=1

b k j (it − P(z2) − t Q(z2, t)) k z2j ,

Im

⎛

⎝(it − P(z2) − t Q(z2, t)) +

∞

k=1

∞

j=0

a k j (it − P(z2) − t Q(z2, t)) k z2j

⎞

⎠

⎞

⎠ ≡ 0,

(7)

or equivalently,

Re

⎛

⎝∞

k=1

∞

j=0

a k j (it − P(z2) − t Q(z2, t)) k z2j

⎞

⎠

+ P

⎛

⎝z2+ ∞

k , j=1

b k j (it − P(z2) − t Q(z2, t)) k z2j

⎞

⎠ − P(z2)

+ t

⎡

⎣Q

⎛

⎝z2+ ∞

k , j=1

b k j (it − P(z2) − t Q(z2, t)) k z2j ,

t+ Im

⎛

⎝∞

k=1

∞

j=0

a k j (it − P(z2) − t Q(z2, t)) k z2j

⎞

⎠

⎞

⎠ − Q(z2, t)

⎤

⎦ + Im

⎛

⎝∞

k=1

∞

j=0

a k j (it − P(z2) − t Q(z2, t)) k z2j

⎞

⎠

× Q

⎛

⎝z2+ ∞

k , j=1

b k j (it − P(z2) − t Q(z2, t)) k z2j ,

t+ Im

⎛

⎝∞

k=1

∞

j=0

a k j (it − P(z2) − t Q(z2, t)) k z2j

⎞

⎠

⎞

on 0 × (−δ0, δ0) for some 0, δ0> 0.

If f1(z1, z2) ≡ z1, then let k1= +∞ In the contrary case, let k1be the smallest integer

k such that a k j = 0 for some j ∈ N∗ Then let j

1 be the smallest integer j such that

a k1j = 0 Similarly, if f2(z1, z2) ≡ z2, then denote by k2= +∞ Otherwise, let k2be the

smallest integer k such that b k j = 0 for some j ∈ N∗ Denote by j2the smallest integer j

such that b k2j = 0

Since P (z2) = o(|z| j ) for any j ∈ N, inserting t = αP(z2) into (8) (withα ∈ R to be

chosen later) one gets

Re



a k1j1P k1(z2)(αi − 1) k1

z j1

2 + o(|z2|j1)

+ Pz2+ b k2j2P k2(z2)(αi − 1) k2



z j2

2 + o(|z2|j2)− P(z2)

+ αP(z2)Q



z2+ b k j P k2(z2)(αi − 1) k2

z j2+ o(|z2|j2),

The second aim of this paper is to show that the stability group of a radially symmetric hypersurface of infinite type inC2< /small>is exactly the one-parameter group< i>{R t}t∈R... paper is organized as follows In Section2, we give several properties of functions vanishing to infinite order at the origin In Section3, we prove TheoremA Section4 is devoted to the proof of. .. of TheoremB Finally, a theorem is pointed out in Appendix

2 Preliminaries

In