On the set of complex points of a 2-sphere

On the set of complex points of a 2-sphere

NIKOLAYSHCHERBINA

Abstract. LetG be a strictly pseudoconvex domain inC²withC^∞-smooth boundary∂G. LetSbe a 2-dimensional sphere embedded into∂G. Denote by E the set of all complex points on S. We study how the structure of the setE depends on the smoothness ofS.

Mathematics Subject Classification (2000):32T15 (primary): 32V40, 53D10 (secondary) .

1.

Introduction

Let M be a 2-dimensionalC¹-smooth manifold in

C

^{2. A point pon M is called} a complex point, if the tangent plane TpM to M at p is a complex line. Denote by

E

the set of all complex points on M. If M is smooth enough and in a general position, then the set

E

consists of isolated points. In this case the topology of M can be described in terms of the local behaviour of M near the points of

E

(see [11]). The structure of the setMnear the points in

E

plays a key role in different questions of complex analysis (see, for example, [2, 3, 9, 10, 12, 15]. In this paper we study the structure of the set

E

in the case, whenM is a 2-dimensional sphere, denoted by S in what follows, embedded into the boundary∂G of aC^∞-smooth strictly pseudoconvex domain G in

C

² (this case is important for applications, as was shown in [5] and [6]). It is easy to see that, due to strict pseudoconvexity of G, the set

E

has no interior points in S. Our goal here is to give a more explicit description of the set

E

depending on the smoothness ofS. Recall, that a manifold is said to be of classC²⁻if it can be represented locally as the graph of a function that belongs to the class Lip^1,α for each positiveα <1. Our main result can now be formulated as follows.

Theorem 1.1. Let G be a strictly pseudoconvex domain in

C

^{2 with C∞-smooth} boundary∂G. Let S be a2-dimensional sphere embedded into∂G. Then, depend- ing on the smoothness of S, the following holds:

1) If S is of class C², then the set

E

of complex points of S is contained in a C¹- smooth nonclosed curveγ ⊂S.

Received January 11, 2008; accepted July 9, 2008.

2) There exists a2-dimensional sphere S ⊂ ∂G of class C²⁻ such that the set

E

contains a Jordan curve of positive 2-dimensional measure.

ACKNOWLEDGEMENTS. Part of this work was done while the author was a visi- tor at the Max Planck Institute of Mathematics (Bonn) and at the Scuola Normale Superiore (Pisa). It is my pleasure to thank these institutions for their hospitality and excellent working conditions. I also express my gratitude to N. G. Kruˇzilin for valuable comments.

2.

Proof of the first part of the theorem

We start with an argument which goes back to Bishop [2] (see also [9]). Namely, if pis a point of

E

, then after a polynomial change of coordinates that movespto the origin we can locally represent Sas the disc

D=

(z, f(z))∈

C

^2:z∈

with being a small disc centered at the origin and f being a complex valued C²-smooth function. Moreover, in view of strict pseudoconvexity ofG, after this change of coordinates the function f will have the special form

f(z)= 1

2|z|²−βRez²+o(|z|²), β≥0

near zero. Recall, that zero is called an elliptic point if 0 ≤ β < ¹₂, a hy- perbolic point if β > ¹₂ and aparabolic point if β = ¹₂. Elliptic and hyper- bolic points are always isolated in

E

. In the case of a parabolic point we can use the real coordinates z = x +i y and represent f as f(z) = y² +o(|z|²). Hence ∂_¯zf(z) = i y+o(|z|) and then, by the implicit function theorem, we ob- tain that σ = {z∈:Im ∂z_¯f(z)=0} is a C¹-smooth curve and locally

E

= {(z, f(z)):z∈σ and Re ∂z_¯f(z)=0}. Therefore, locally the set

E

is a closed sub- set of aC¹-smooth curve.

Since the set

E

is compact, the only obstruction for

E

to be a subset of a non- closedC¹-smooth curveγ ⊂ Sis that there is a closedC¹-smooth curve⊂

E

^.

Assume, to get a contradiction, that such a closed C¹-smooth curve ⊂

E

exists. Consider a complex tangentialC^∞-smooth closed curvein∂G(complex tangential here means that for each point p∈the tangent lineT_ptoat pis contained in the complex tangent planeT_p^C(∂G)to∂G at p) close enough toin theC¹-metric. Then, using a partition of unity along the curve, we can construct a smallC¹-perturbationSof Sin∂Gsuch that ⊂ S and each point inis a complex point onS. Moreover,Scan be madeC^∞-smooth in a neighbourhood of . For each point p ∈ , consider the unit vectoru(p)tangent to, the vector

iu(p) ∈ T_p^C(∂G)and the unit vectorn(p) ∈ T_p(∂G)orthogonal to T_p^C(∂G)and such that the vectors(u(p),iu(p),n(p))define the positive orientation of ∂G at the point p. Let O(p)be the rotation of T_p(∂G)around the direction u(p)that transforms the vectoriu(p)into the vectorn(p). Using the tubular neighbourhood theorem (see, for example, [8, Theorem 1.4]) we can changeSin a neighbourhood of to get a new 2-sphere,S ⊂∂G,C^∞-smooth nearsuch that ⊂ S and

n(p)∈ T_p(S)for each point p ∈ . It is easy to see that S is totally real near . Then we can perturbS slightly outside a small neighbourhood of to get a C^∞-smooth 2-sphere S˜ ⊂ ∂G in general position. To finish the proof of the first part of our theorem we use an argument of Gromov (see [7, page 342]). Namely, by the result of Bedford-Klingenberg [3] and Kruˇzilin [10], there is a smooth 3-ball

B

which is the disjoint union of holomorphic discs{D_α}, such that∂

B

= ˜S. By Chirka’s theorem [4] we know that discs D_αareC^∞-smooth to the boundary∂D_α near the totally real part of S˜ (i.e. outside of finitely many complex points of S)˜ and, moreover, the boundary∂D_αof each disc D_α isC^∞-smooth at this part ofS˜ and transversal there to the distribution {T_p^C(∂G)} of complex tangencies to∂G.

Consider a disc D_α0from the family{D_α}such that its boundary∂D_α0 “touches”

the curve“for the first time” and let pbe a point of ∂D_α0∩. More precisely, letD_α0 ⊂

B

be a holomorphic disc with the property that∂D_α0∩ = ∅and such that for some connected component of the set

B

^\Dα0, each holomorphic discD_α, which is contained in this component, satisfies∂D_α∩= ∅. Now we can see that, on the one hand, since the curves∂D_α0andare tangent to each other at the point p, and since the curve was chosen to be complex tangential, the curve∂D_α0 is complex tangential at p. On the other hand, since the point p is contained in the totally real part of S, the curve˜ ∂D_α0 has to be transversal toT_p^C(∂G). This gives the desired contradiction and completes the proof of the first part of the theorem.

Remark 2.1. In the special case when the boundary of the domainGis diffeomor- phic to a 3-dimensional sphere, the fact that the closed complex tangential curve ⊂ S⊂∂Gmentioned above does not exist can also be deduced from the [1, The- orem 1].

3.

Proof of the second part of the theorem

We prove the second part of the theorem in several steps. First, we construct a spe- cial arc E⊂

R

²x,yof positive 2-dimensional measure. Then we define a functionG on Esuch that G ∈C²⁻(E) with the functionsG_x(x,y)= yandG_y(x,y)= 0 chosen to be the first derivatives ofGonE. Next, following an idea of H. Whitney (see [14]), we construct a nonconstant function H ∈ C²⁻(E)with H_x(x,y) = 0 and H_y(x,y) = 0. Using G and H we define a function F ∈ C²⁻(E) with F_x(x,y) = y and F_y(x,y) = 0 which is zero at both endpoints of E. Then, using E, we construct a Jordan curve E˜ ⊂

R

²x,y of positive 2-dimensional mea- sure and, using F, we define a functionF˜ onE˜ of classC²⁻(E˜)with derivatives

F˜_x(x,y) = y and F˜_y(x,y) = 0. Next, applying Whitney’s extension theorem to the function F, we contruct a 2-sphere˜ S² ⊂

R

³x,y,z of classC²⁻ which contains a Jordan curve of positive 2-dimensional measure such that at each point of this curve the tangent plane to S² coincides with the corresponding plane of the stan- dard contact distribution in

R

³x,y,z. Finally, using the Darboux theorem, we embed this sphere into the boundarybGof the given strictly pseudoconvex domainG.

3.1. Construction of the arc E

First, we define for eachα∈(0,1)an auxiliary set

E

^α=

0,1−α 2

∪

1+α 2 ,1

×

0,1−α 2

∪ 1+α

2 ,1

∪({0} × [0,1])∪

[0,1] ×

1+α 2

∪({1} × [0,1]) .

We denote A = (0,0),B = (1,0),Q₀ = [0,^1−α₂ ] × [0,^1−α₂ ],Q₁ = [0,^1−α₂ ] × [^1+α₂ ,1],Q₂ = [^1+α₂ ,1] × [^1+α₂ ,1], and Q₃ = [^1+α₂ ,1] × [0,^1−α₂ ]. Further, we denote A₀ = A=(0,0),B₀ =(0,^1−α₂ ),A₁= (0,^1+α₂ ),B₁=(^1−α₂ ,^1+α₂ ),A₂ = (^1+α₂ ,^1+α₂ ),B₂ =(1,^1+α₂ ),A₃ = (1,^1−α₂ ), and B₃ = B = (1,0)(see the set

E

^α in Figure 3.1).

Q₁ Q₂

Q₀ Q₃

B₁ A₂ B₂

A₁

A₀= A = (0, 0) B₃= B = (1, 0)

A₃ B₀

α

α

Figure 3.1.The setE^α.

To define the set E we consider the sequenceαn = _(n+¹₁₎2,n = 1,2, . . .We con- struct the set E as the intersection of a decreasing sequence of compact sets E_n which will be defined inductively. We set E₁ =

E

^α1. To define the set E₂ we consider the image

E

˜^{α2 of the set}

E

^α2 under the homothety with coefficient ^1−α₂¹.

Then for eachi = 0,1,2,3 we consider the set

E

ⁱ obtained from the set

E

˜^{α2 by} an orthogonal transformation (if necessary) and translation in such a way that the image of the points Aand Bwill coincide with the points Ai andBi, respectively.

It is easy to see that we need an orthogonal transformation only fori = 0,3. The set E₂is obtained from the set E₁by replacing for eachi = 0,1,2,3 the square Q_i by the corresponding set

E

^{i. For each j =0,1,2,}3 we denote byQ_{i j},A_{i j} and B_{i j}the images of the square Q_j and the points A_j,B_j in the corresponding set

E

^i, respectively (the setE₂is shown in Figure 3.2).

Q₁₁ Q₁₂ Q₂₁ Q₂₂

Q₁₀ Q₁₃ Q₂₀ Q₂₃

Q₀₃ Q₀₂ Q₃₁ Q₃₀

Q₀₀ Q₀₁ Q₃₂ Q₃₃

B₃₁ A₀₂

A₃₂ B₀₁

B₁₂

A₁₁ A₂₁

A₁₃ B₁₀

B₂₂ A₂₃ B₂₀

A₁₀ = A₁ B₂₃ = B₂

B₀₃ = B₀ A₃₀ = A₃

A₀₀ = A₀ = A B₃₃ = B₃ = B

B₁₃ = B₁ A₂₀ = A₂

A₀₃ B₀₂ A₃₁ B₃₀ B₁₁ A₁₂ B₂₁ A₂₂

B₀₀ A₀₁ B₃₂ A₃₃ Figure 3.2.The setE2.

To describe the inductive step of our construction we assume that the set E_n is al- ready constructed and define the set E_n+1. Consider the image

E

˜^{αn+1 of the set}

E

^αn+1under the homothety with coefficient _iⁿ_=11−αi

2

. Then for each multiindex (i₁,i₂, . . . ,i_n),i_j = 0,1,2,3, j = 1,2, . . . ,n, consider the set

E

ⁱ1...in obtained from the set

E

˜^αn+1by an orthogonal transformation (if necessary) and translation in such a way that the image of the pointsAandBwill coincide with the pointsA_i1...in and B_i1...in, respectively. The set E_n+1 is obtained from the set E_n by replacing each square Q_i1...in by the corresponding set

E

ⁱ1...i_n. For eachi_j =0,1,2,3, j = 1,2, . . . ,n +1, we denote by Q_i1...in+1,A_i1...in+1 and B_i1...in+1 the images of the square Q_in+1 and the points A_in+1,B_in+1 in the corresponding set

E

ⁱ1...i_n, respec- tively. Note, that for each multiindex(i₁, . . . ,i_n)one has A_i1...in0 = A_i1...in and

B_i1...in3=B_i1...in.

Since {E_n} is a decreasing sequence of compact sets, E = _∞

n=1E_n is a nonempty compact subset of

R

²x,y. It is easy to see that it is an arc (see the set Ein Figure 3.3).

Figure 3.3.The setE.

To estimate the area of the setEwe observe that Area(E_n)=(1−αn)² Area(E_n−1)=

n k=1

(1−αk)²= n k=1

1− 1

(k+1)² 2

= 1

2

1+ 1 n+1

2

> 1 4

for everyn=1,2, . . .. Hence, the setEhas a positive 2-dimensional measure (and, moreover, Area (E)= ¹₄).

3.2. Definition and properties of the functionG

For eachn=1,2, . . .letnbe the connected component of the set(0,1)×(−1,1)\

E_ncontaining the square(0,1)×(−1,0)and let J_n=∂n∩([0,1] × [0,1]). On each curve J_n we define a functionG_n in the following way. For a point p ∈ J_n we denote by J_n^p a part of J_n with initial point Aand endpoint pand then we set G_n(p)=

J_n^pyd x.

We will need the following estimate on the functionG_n.

Lemma 3.1. Let Q_i1i2...im be a square obtained after m≥6steps of our construc- tion above. Then for each natural number n >m and all points p,q ∈J_n∩Q_i1i2...im one has

G_n(q)−G_n(p)−

[p,q]yd x

<Area(Q_i1i2...im)= 1 2^2m+2

m+2 m+1

₂

. (3.1) Proof. We proceed by induction onmhaving the numbernfixed and decreasing the numberm. For each pair of pointsr,s∈ J_nwe denote byJ_n^r,sthe part of the curve

J_nwith the endpointsrands. Form=n−1 we observe from the definition of the functionG_n and Green’s theorem thatG_n(q)−G_n(p)−

[p,q]yd xrepresents the sum (with signes) of the areas of domains bounded by the arc J_n^p,qand the segment [p,q](see Figure 3.4).

Q_{i1i2…in−1} J_n^p,q

p

q

Figure 3.4.The curveJn^p,qin the squareQi₁i₂...i_n−1.

Since these domains are contained in the square Q_{i1i2...in−1}, we conclude that the inequality (3.1) holds true in this case. The same argument cannot be applied in the case of generalmdue to the fact that the corresponding domains will not be disjoint anymore and, therefore, their areas will be counted with multiplicities. That is why in the case of an arbitrarymwe need to improve the argument above and to use the inductive procedure.

Assume that the inequality (3.1) holds true for all squares Q_i1i2...im and all points p,q ∈ J_n∩ Q_i1i2...im withm ≥ m₀+1 (nbeing fixed). Consider a square Q_i1i2...im

0 and points p,q ∈ J_n∩ Q_i1i2...im

0. Observe first that the segment[p,q] can intersect at most three of the squares Q_i1i2...im

0im0+1, i_m0+1 = 0,1,2,3. We assume in what follows that it intersects exactly three of them (in the other cases the argument is easier, since some terms in our estimates will disappear), namely the squares Q_i1i2...im

00,Q_i1i2...im

01 and Q_i1i2...im

02 (the cases of another three squares can be treated similarly). Consider an orientation of the segment[p,q] from the point pto the pointqand for eachi =0,1,2 denote byp_i and p_i the first and the last, respectively, (according to the orientation of[p,q]) points of the set[p,q] ∩

J_n^p,q∩ Q_i1i2...im

0i. Note, that p₀= pand p₂=q. For eachi =0,1,2 we have by our inductive hypotesis that

G_n(p_i)−G_n(p_i)−

[p_i,p_i]yd x

< Area(Q_i1i2...im

0i)< 1

4Area(Q_i1i2...im

0). (3.2) Since the segment[p₀,p₁] intersects the curve J_n only at its endpoints, we con- clude from the Green’s theorem thatG_n(p₁)−G_n(p₀)−

[p₀,p₁]yd xrepresents the

area of the domain ₁bounded by the curve J^p

0,p₁

n and the segment[p₀,p₁](see Figure 3.5).

q = p'₂ p₂ p'₁

p₁

p = p₀ p'₀

Ω^i1i2…im0 Σ1

Figure 3.5.The sets 1andⁱn^1i2...im0 in the squarei₁i₂...i_m0.

Since the domain ₁is contained in the setⁱn^1i2...im0 =n∩Q_i1i2...im

0, and since n∩Q_i1i2...im

0 ⊂ Q_i1i2...im

0\ E_n, it follows that 0<G_n(p₁)−G_n(p₀)−

[p₀,p₁]yd x=Area( 1)<Area(n∩Q_i1i2...im

0)

< Area(Q_i1i2...im

0)− Area(E_n∩Q_i1,i2...im

0)

= Area(Q_i1i2...im

0)(1−(1−αm₀+1)². . . (1−αn)²)

=Area(Q_i1i2...im

0)

1−

1− 1

(m₀+2)² 2

. . .

1− 1

(n+1)² 2

=Area(Q_i1i2...im

0)

1−

(m₀+1)(n+2) (m₀+2)(n+1)

2

< Area(Q_i1i2...im

0)

1−

m₀+1 m₀+2

2

< 2

m₀+2 Area(Q_i1i2...im

0).

(3.3)

Similarly, G_n(p₂)−G_n(p₁)−

[p₁,p₂]yd x represents the area with the negative sign (due to the orientation of its boundary) of the domain ₂bounded by the curve

J^p

1,p₂

n and the segment [p₁,p₂]. Let be the smallest rectangle in Q_i1i2...im that coutains the squaresQ_i1i2...im 0

012,Q_i1i2...im

013,Q_i1i2...im

020andQ_i1i2...im

021(see Figure 3.6).

q = p'₂ p₂

p'1

p₁

p = p₀ p'0

Σ₂ Π

Figure 3.6.The sets 2andin the squarei₁i₂...i_m

0.

Since the domain ₂is a subdomain of, and since we have an obvious estimate Area() < ¹₄ Area(Q_i1i2...im

0), we conclude that 0>G_n(p₂)−G_n(p₁)−

[p₁,p₂]yd x = −Area( 2) >−Area()

>−1

4Area(Q_i1i2...im

0).

(3.4)

Taking into account different signs of the terms in the inequalities (3.3) and (3.4) we have form₀≥6 the following estimate

G_n(p₁)−G_n(p₀)−

[p₀,p₁]yd x

+

G_n(p₂)−G_n(p₁)−

[p₁,p₂]yd x

< 1

4Area(Q_i1i2...im

0).

(3.5)

Finally, the inequalites (3.2) and (3.5) imply that G_n(q)−G_n(p)−

[p,q]yd x <

2 i=0

(G_n(p_i)−G_n(p_i)−

[pi,p_i]yd x +

G_n(p₁)−G_n(p₀)−

[p₀,p₁]yd x

+(G_n(p₂)−G_n(p₁)−

[p₁,p₂]yd x

< 3

4Area(Q_i1i2...im

0)+ 1

4 Area(Q_i1i2...im

0)=Area(Q_i1i2...im

0).

This completes the proof of the lemma.

Now we define the functionGon the arcE. Letpbe a point ofE. Consider a sequence of points p_n ∈ J_n,n = 1,2, . . ., such that p_n → p, asn → ∞and set G(p)=limn→∞Gn(pn).

To prove that the functionGis well defined we consider the Cantor set

Q

^def= _∞

n=1

(i₁,i₂,...,i_n)Q_i1i2...in and observe that since the set E\

Q

is constituted by horizontal and vertical segments, and in view of the definition of the functions G_n, it is enough to prove that the sequence G_n(p_n)converges for p_n → p with p,p_n ∈

Q

^{, n=1,2, . . .Let pn}1 andp_n2,n₁ >n₂, be two points of our sequence p_n ∈ J_n

Q

^{, n = 1,2, . . ., and let Qi}1i₂...i_m be the smallest of the described above squares that contains both of these points. Observe that the vertex A_i1i2...im of the squareQ_i1i2...im is contained in both curvesJ_n1andJ_n2. It follows then from Green’s theorem thatG_n1(A_i1i2...im)−G_n2(A_i1i2...im)represents the area of the set bounded by the curves J_n^A₁^,Ai1i2...im and J_n^A₂^,Ai1i2...im. Since this set is a subset of the setn₁\ ¯n₂, we conclude that

|G_n1(A_i1i2...im)−G_n2(A_i1i2...im)|< Area(n₁\ ¯n₂). (3.6) For each j =1,2 we have (by the lemma):

G_nj(p_nj)−G_nj(A_i1i2...im)−

[A_i1i2...im,p_{n j}]yd x

< Area(Q_i1i2...im).

If we denote byl_mthe length of a side ofQ_i1i2...im (it depends only onm), then the last inequality gives us

2 j=1

|G_nj(p_nj)−G_nj(A_i1i2...im)|<

2 j=1

[Ai1i2...im,p_{n j}]yd x

+2 Area(Q_i1i2...im)

<2l_m+2 Area(Q_i1i2...im) <3l_m=3 m i=1

1−αi

2

= 3

2^m+1 ·m+2

m+1. (3.7) It follows now from (3.6) and (3.7) that

|G_n1(p_n1)−G_n2(p_n2)|≤

2 j=1

|G_nj(p_nj)−G_nj(A_i1i2...im)|

+ |G_n1(A_i1i2...im)−G_n2(A_i1i2...im)|< 3

2^m+1 · m+2

m+1 +Area(n1\ ¯n2).

Since for points p,p_n1,p_n2 ∈

Q

^{the numberm}will tend to infinity as p_n1,p_n2 → p, and since Area(n₁\ ¯n₂) → 0 asn₁,n₂ → ∞, we conclude that the limit of the sequence G_n(p_n)asn → ∞ exists and, therefore, the functionG is well defined.

Next, we prove thatG ∈C²⁻(E)withG_x(x,y)= yandG_y(x,y)=0 chosen to be the first derivatives of G at each point(x,y) ∈ E. To do this rigorously we recall the definition of a function belonging to the classC²⁻(E)(this definition is due to H. Whitney. Further details can be found, for example, in [13]).

Definition 3.2. LetEbe a compact subset of

R

²x,y and let f be a function defined onE. We say that f belongs to the classC²⁻(E)if there exist bounded functions f_x and f_y defined onEwith the property that for eachε >0 there is a constantM such that

|f(x+x,y+y)− f(x,y)− f_x(x,y)x− f_y(x,y)y|

≤M(|x| + |y|)^2−ε (3.8) for all(x,y), (x+x,y+y)∈E.

To prove thatG∈C²⁻(E)we consider two points p,p+p∈E. Since the functionG obviously is smooth on each of the segments[B_i1...in,A_i1...(in+1)],i_n = 0,1,2,and satisfies condition (3.8) withG_x(x,y) = y andG_y(x,y) = 0 there, the general case, when p,p+ p ∈ E, can be easily reduced to the case when pand p+ pare contained in the Cantor set

Q

. That is why we assume in what follows that p,p+ p∈

Q

. Consider as above a numberm such thatp,p+p belong to a squareQ_i1...im for some indices(i₁, . . . ,i_m), but not to a smaller square Q_i1,...imim+1,i_m+1 = 0,1,2,3. Since p and p+pbelong to different squares Q_i1...imim+1and Q_i1...imi

m+1, it follows that the distance between these points is not less than the minimal distance between Q_i1...imim+1andQ_i1...imi

m+1, that is,

|p| ≥αm+1

1−α1

2

. . .

1−αm

2

= 1

(m+2)² · 1 2^m

m k=1

1− 1

(k+1)²

= 1

2^m+1 · 1

(m+1)(m+2). (3.9) Now we estimate the left hand side of the condition (3.8) for our functionG

L

G(p,p+p)^def= G(p+p)−G(p)−G_x(p)x−G_y(p)y

=G(p+p)−G(p)−yx, where p=(x,y)and p=(x, y). It is easy to see that

[p,p+p]yd x =yx+1 2xy, hence

|

L

^{G(p,p+p)| ≤G(p+p)−G(p)−}

[p,p+p]yd x +1

2|x||y|. (3.10)