On holomorphic curves tangent to real hypersurfaces of infinite type

The purpose of this paper is to investigate the geometric properties of real hypersurfaces of D'Angelo infinite type in ${\mathbb C}^n$. In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether there exists a nonconstant holomorphic curve tangent to a given hypersurface to infinite order. A sufficient condition for this existence is given by using Newton polyhedra,which is an important concept in singularity theory. More precisely,equivalence conditions are given in the case of some model hypersurfaces.


Introduction
Let M be a (C ∞ smooth) real hypersurface in C n and let p lie on M. Let r be a local defining function for M near p (∇r = 0 when r = 0). In [6], [7], the following invariant is introduced: where Γ denotes the set of (germs of) nonconstant holomorphic mappings γ : (C, 0) → (C n , p). (For a C ∞ mapping h : C → C or C n such that h(0) = 0, let ord(h) denote the order of vanishing of h at 0.) The invariant ∆ 1 (M, p) is called the D'Angelo type of M at p. We say that M is of finite type at p if ∆ 1 (M, p) < ∞ and of infinite type at p otherwise (the latter case will be denoted by ∆ 1 (M, p) = ∞). The class of finite type plays crucial roles in the study of the local regularity in the∂-Neumann problem over pseudoconvex domains Ω with smooth boundary ∂Ω. Indeed, it was shown by D. Catlin [4], [5] that M = ∂Ω is of finite type at p if and only if a local subelliptic estimate at p holds. From its importance, real hypersurfaces of finite type have been deeply investigated from various points of view.
On the other hand, to understand the geometric properties of real hypersurfaces of infinite type is also an interesting subject in the study of several complex variables. These hypersurfaces contain some kind of strong flatness. In order to describe the geometric structure of this flatness, the situation of contact of holomorphic curves with the respective hypersurface must be carefully observed. In this paper, we mainly consider the following question: Question 1. When does there exist a nonconstant holomorphic curve γ ∞ tangent to M at p to infinite order?
Since the condition of the desired curve γ ∞ in Question 1 can be written as as t ∈ C → 0, for every N ∈ N, the condition ∆ 1 (M, p) = ∞ is necessary for the existence of the curve γ ∞ . It has been shown in [20], [7], [11] that when M is real analytic, the above two conditions are equivalent (in this case, the curve γ ∞ is contained in M). Moreover, in the case of smooth M, this equivalence is also shown in the formal series sense in [8], [16]. But, in general, the sufficient direction is not true. Indeed, the nonexistense of the curve γ ∞ in (1.2) is shown in the case of some real hypersurfaces constructed in [3], [16], [23], [9]. Understanding flatness on hypersurfaces of infinite type has been recognized to be a delicate issue.
In this paper, in order to investigate the flatness of real hypersurfaces, we use not only holomorphic curves but also Newton polyhedra of defining functions, which plays important roles in singularity theory (cf. [1], [2]). Approach from the viewpoint of singularity theory is useful in the study of types and there have been many works of the sort ( [21], [13], [10], [11], [15], etc.).
Proposition 1.1. Let us consider the following eight conditions for a real hypersurface M at p: (2) ∆ reg 1 (M, p) = ∞; (3) There exists a γ ∈ Γ reg tangent to M at p to infinite order; (4) There exists a γ ∈ Γ tangent to M at p to infinite order; (5) There exists a holomorphic coordinate (z) = (z 1 , . . . , z n ) at p such that p = 0 and a defining function r for M on (z) is not convenient (see Section 2); (6) There exists a holomorphic coordinate (z) at p such that p = 0 and the Newton polyhedron of a defining function for M on (z) (see Section 2) takes the form N + (r) = {(ξ 1 , . . . , ξ n ) ∈ R n + : ξ n ≥ 1}; (7) The Bloom-Graham type of M at p is infinity (i.e. there are complex submanifolds of codimension one tangent to M at p to arbitrarily higher order); Then, among the above eight conditions, the following implications hold: d l P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P (8).
The proof of the above proposition will be given in Section 4.1.
Remark 1.2. In the above proposition, for each implication with only one direction, its opposite direction is not true (see Remark 4.2 in Section 4 for details).
The following theorem gives a sufficient condition for the existence of the curve γ ∞ in Question 1. This condition is described by using Newton polyhedra of defining functions for real hypersurfaces. The definition of N -canonical coordinates will be given in Section 2 (Definition 2.3). The proof of the above theorem will be given in Section 4.2.
The following corollary can be directly obtained from Theorem 1.3.
If there is no γ ∈ Γ tangent to M at p to infinite order, then M does not admit any N -canonical coordinates at p.
It is easy to check that the examples of hypersurface constructed in [3], [16], [23], [9] do not admit any N -canonical coordinates. More exactly, we will give equivalence conditions in more restricted cases in Sections 5 and 6.
Next, it is seen in [15] that when M is the boundary of pseudoconvex Reinhardt domains, M always admits an N -canonical coordinate. Therefore, Theorem 1.3 implies the following. This paper is organized as follows. In Section 2, we recall the concepts: Newton polyhedra, N -nondegeneracy condition and N -canonical coordinates, which were introduced in [15]. In Section 3, for the analysis later, we prepare appropriate coordinates on which hypersurfaces are expressed in clear form. Section 4 is devoted to the proof of results given in the Introduction. More precise results are given in the two dimensional case in Section 5 and the higher dimensional case under the Bloom-Graham infinity type assumption in Section 6. Since the Bloom-Graham type is the same as the D'Angelo type in the two-dimesional case, some results in Section 5 can be considered as special cases of those in Section 6. But, they are separately explained to make clear their difference. Lastly, we consider open problems in Section 7.
Notation, symbols and terminology.
• We always take a good parametrization for corves γ ∈ Γ without any mentioning. That is to say, a point on the curve, defined by t → γ(t), corresponds to only one value of t. For example, (t, t 2 ) is good, but (t 2 , t 4 ) is not good. • We use the words pure terms for any harmonic polynomial and mixed terms for any sum of monomials that are neither holomorphic nor anti-holomorphic.

Newton polyhedra for real hypersurfaces
Let us define the Newton polyhedron of a real-valued smooth function F defined near the origin in C n . The Taylor series expansion of F at the origin is The Newton polyhedron of F is defined by where S(F ) = {α + β ∈ Z n + : C αβ = 0}. The Newton diagram N (F ) of F is defined to be the union of the compact faces of N + (F ). We use coordinates (ξ) = (ξ 1 , . . . , ξ n ) for points in the plane containing the Newton polyhedron. The following classes of functions F simply characterized by using their Newton polyhedra often appear in this paper: • F is called to be flat at 0 if N + (F ) is an empty set.
• F is called to be convenient at 0 if N + (F ) meets every coordinate axis. Let (z) = (z 1 , . . . , z n ) be a holomorphic coordinate around p such that p = 0. Let r be a local defining function for M near p on the coordinate (z). For a given tuple (M, p; (z)), we define a quantity ρ 1 (M, p; (z)) ∈ Z + ∪ {∞} as follows. If r is convenient, then let ρ 1 (M, p; (z)) := max{ρ j (r) : j = 1, . . . , n}, where ρ j (r) is the coordinate of the point at which the Newton diagram N (r) intersects the ξ j -axis. Otherwise, let ρ 1 (M, p; (z)) := ∞. We remark that ρ 1 (M, p; (z)) depends on the chosen coordinate (z), but it is independent of the choice of defining functions after fixing a coordinate. Considering the curves γ j (t) = (0, . . . , (j) t , . . . , 0) ∈ Γ reg for j = 1, . . . , n, we can see that the inequality ρ 1 (M, p; (z)) ≤ ∆ reg 1 (M, p) always holds. Next, let us introduce an important concept "N -nondegeneracy condition" on a smooth function F defined near the origin in C n .
Let κ be a compact face of N + (F ). The κ-part of F is the polynomial defined by The set of holomorphic curves Γ κ is defined by where c = (c 1 , . . . , c n ) ∈ (C \ {0}) n , a = (a 1 , . . . , a n ) ∈ N n and "a ∈ N n determines κ" means that the set {ξ ∈ N + (F ) : n j=1 a j ξ j = l} coincides with the face κ for some l ∈ N.
The above concept is analogous to the nondegeneracy condition introduced by Kouchnirenko [19], which plays important roles in the singularity theory. Detailed properties of this condition are explained in [15].
The following relationship ) is always established for every coordinate (z) at p. The following theorem shows that the equalities in (2.3) are satisfied under the N -nondegeneracy condition.

Theorem 2.3 ([15]
). If there exists an N -canonical coordinate (z) at p, then the following equalities hold: ). Note that the above theorem is valid for the infinite type case. From the above theorem, the existence of N -canonical coordinates implies that both values of ∆ 1 (M, p) and ∆ reg 1 (M, p) can be directly seen from geometrical Newton data of M at p.

Standard coordinates
Let M be a real hypersurface in C n+1 (n ≥ 1) and let p lie in M.
It follows from Taylor's formula that there exists a holomorphic coordinate (z, w) := (z 1 , . . . , z n , w) at p on which a local defining function r for M is expressed near the origin as in the following form: Of course, there may be many such coordinates, which are said to be standard for M at p.
Furthermore, if there exists a holomorphic coordinate (z, w) := (z 1 , . . . , z n , w) around p on which a local defining function r for M is expressed near the origin as in the model form: where F is as in (3.1), then (z, w) is called a good (standard) coordinate for M at p. Note that good coordinates do not always exist for all hypersurfaces.

Proofs of results in the Introduction
Let M be a real hypersurface in C n and let p lie in M.  (iii) There exists k ∈ {1, . . . , n} such that the Taylor series of r does not contain any term consisting of z k andz k only; The following lemma expresses a property of flat hypersurfaces by using the language of Newton polyhedra.  (7) is easy to see from Lemma 4.2. (7) =⇒ (2) is shown in [9]. (Lemma 5 in [9] states (7) =⇒ (1), but its proof actually implies the above stronger implication.) The other implications are obvious.  .2). Thus, we only write a function F (z,z) for each case. In the two-dimensional case, the converses of some implications are also true (see Section 5). In these cases, counterexamples must be constructed in C n with n ≥ 3.
It is shown in [16], [23] (see also Corollary 6.4 in this paper) that the second example shows (2) ⇒ (3) and (1) ⇒ (4) in the two-dimensional case. The higher dimensional case can be easily shown. It is easy to check that the other hypersurfaces are counterexamples.
Let (z) be an N -canonical coordinate for M at p and let r be a local defining function for M near p on the coordinate (z). From Theorem 2.3, the condition (1) implies ρ 1 (M, p; (z)) = ∞, which is equivalent to the condition (5) from Lemma 4.1.

Two dimensional case
In this section, we more precisely consider Question 1 in the Introduction in the case when a real hypersurface M is in C 2 . Let p ∈ M.
In the two dimensional case, many implications in Proposition 1.1 can be refined by equivalences.
Proposition 5.1. Let M be a real hypersurface in C 2 and let p lie in M. Among the above eight conditions for M at p in Proposition 1.1, the following implications hold: d l P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P (8). (6) is obvious. It is known in [7] that (1), (2), (7) are equivalent. (4) =⇒ (3) will be shown in Lemma 5.2, below.
Lemma 5.2. If γ ∈ Γ is tangent to M ⊂ C 2 at p to infinite order, then γ ∈ Γ reg .
Proof. Let (z, w) be a standard coordinate for M at p such that M is expressed as in (3.1).
First, if F is flat at 0, then the curve γ, satisfying (r •γ)(t) = O(t N ) for every N ∈ N, essentially takes the form: γ(t) = (t, 0). This curve is regular.
Next, let us consider the case when ord(F ) = m with some m ∈ N. In this case, the Newton polyhedron of r takes the form N + (r) = {ξ ∈ R 2 + : ξ 1 /m + ξ 2 ≥ 1}. Let κ := {ξ ∈ R 2 + : ξ 1 /m + ξ 2 = 1}, which is the only compact facet of N + (r). If r κ were N -nondegenerate, then Theorem 2.3 implies ∆ 1 (M, p) = m < ∞, which is a contradiction. If r κ is not N -nondegenerate, then a desired curve must be written in the form γ(t) = (t, ct m + O(t m+1 )) with c = 0 from the definition of the N -nondegeneracy. This curve is also regular.
From Theorem 2.3, we can see the following. The following lemma is essentially the same as Theorem 2 in [16] (see also [18]). Proof. Let us assume the existence of a standard coordinate (z, w) on which the Taylor series of F contains a mixed term. Let m be the minimum order of the mixed terms of Taylor series of F and let p m (z,z) denote the sum of the mixed terms of order m.
It is easy to construct a standard coordinate (z, w * ) on which M can be expressed by the equation: with ord(Q) ≥ m + 1 and R * 1 , R * 2 have the same properties as those of R 1 , R 2 in (3.1), respectively. Since 2Re(w * ) + p m (z) is N -nondegenerate, the above (z, w * ) is canonical for M at p. It follows from Theorem 2.3 that ∆ 1 (M, p) = m < ∞, which is a contradiction.
From the above lemma, when ∆ 1 (M, p) = ∞, the Taylor series of F can be expressed as 2 ∞ j=2 Re(c j z j ) where c j ∈ C. When the hypersurface M and p ∈ M are fixed, the sequence of complex numbers {c j } j∈N is determined by the chosen standard coordinates (z, w) only. For a given coordinate (z, w), we define the formal power series: If M is real analytic, then the series S(z) converges near the origin. But, its converse is not always true. From Cauchy-Hadamard's formula, positivity of the convergence radius of the power series S(z) is equivalent to the condition lim sup j→∞ |c j | 1/j < ∞.
Hereafter in this section, we only consider the case when the hypersurface admits a good coordinate at p. That is to say, M can be expressed as in the model form where F is the same as that in (3.1).
The following theorem gives equivalence conditions for Question 1 in the Introduction in the two-dimensional model case. First, let us show (iii) =⇒ (ii). From (iii), the power series S(z) can be regarded as a holomorphic function defined on some open neighborhood of z = 0. Putting w * = w − S(z), we can express the hypersurface M on the good coordinate (z, w * ) by the equation 2Re(w * ) + F * (z,z) = 0, where F * (z,z) = F (z,z) − 2Re(S(z)). Since F (z) − 2Re(S(z)) is flat at z = 0, the existence of coordinate in (ii) is shown.
Next, let us show (i) =⇒ (iii). We may assume that the regular holomorphic curve in (i) can be expressed as γ(t) = (t, −h(t)) on a good coordinates (z, w) satisfying that Here h is holomorphic near the origin and satisfies h(0) = 0. Let ∞ j=1 a j t j be the Taylor series of h at t = 0, which converges on an open neighborhood of t = 0. Now, after the first finite sum of the Taylor series of h and F are substituted into the equations (r for every N ∈ N. From the above equality, we can see that c j = a j for every j ∈ N. This means that S(t) converges and S(t) = h(t) on an open neighborhood of t = 0.
Remark 5.6. The condition (iii) in Theorem 5.5 is weaker than the condition: M is real analytic near p. Indeed, as mentioned in the Introduction, the real analyticity of M implies the existence of desired curves in (i) in the theorem ( [20], [7], [11]). The hypersurface M defined by 2Re(w) + e −1/|z| 2 = 0 with p = (0, 0) satisfies the conditions (ii) and (iii), but M is not real analytic at p.
In [3], [16], [23], smooth real hypersurfaces in C 2 admitting no curve γ ∞ in (1.2) are constructed. It follows from Theorem 5.5 that there are many such hypersurfaces. (i) There is no γ ∈ Γ tangent to M at 0 to infinite order; (ii) The convergence radius of the power series S(z) equals zero.
Remark 5.8. From (ii) in the above corollary, it is quite easy to construct smooth hypersurfaces satisfying the condition (i). Let {c j } j∈N be a sequence of complex numbers such that the convergence radius of ∞ j=2 c j z j is zero. By using a classical lemma of E. Borel (cf. [22], Theorem 1.5.4, or [14], Theorem 1.2.6), for the formal power series of (x, y) ∈ R 2 (z = x + iy): there exists a real-valued C ∞ function f defined near the origin in R 2 whose Taylor series at the origin is (5.3). Then the real hypersurface defined by 2Re(w) + f (z,z) = 0 satisfies the condition (i). We remark that the above hypersurface cannot always be uniquely determined from the sequence {c j } j∈N .
In [16], [23], similar examples of hypersurfaces have been found. Since their constructions do not use a lemma of E. Borel directly, they look much more elaborate.
Furthermore, it is shown in [9] that when {c j } j∈N is an increasing sequence of positive real numbers, the above function f can be selected to be a subharmonic function on C. Therefore, there are many pseudoconvex real hypersurfaces satisfying the condition (i) in the corollary.

Higher dimensional case
In this section, we generalize results given in the previous section to the higher dimensional case. Let M be a real hypersurface in C n+1 (n ≥ 1) and let p lie in M. In this section, we always consider the case when M is of Bloom-Graham infinite type at p. First, let us recall the exact definition of the Bloom-Graham type in [3]. We remark that the following definition is an equivalence condition for their original type. This equivalence is also shown in [3]. Definition 6.1. Let X be a set of n-dimensional complex submanifolds containing p. We say that the Bloom-Graham type of M is m (< ∞), if there is a X ∈ X tangent to M at p to order m but no X ∈ X tangent to a higher order. Otherwise, we say that the Bloom-Graham type of M is infinity at p (see the condition (7) in Proposition 1.1).
In the case of Bloom-Graham infinity type, a similar property to Lemma 5.4 can be seen. Proof. Let us assume the existence of a standard coordinate (z, w) on which the Taylor series of F contains a mixed term. Let m be the minimum order of the mixed terms of Taylor series of F .
It is easy to construct a new standard coordinate (z, w * ) on which M can be expressed by the equation 2Re(w * ) + P m (z,z) + Q(z,z) + R * 1 · Im(w * ) + R * 2 = 0, where P m is a non-zero mixed homogenous polynomial of degree m without pure terms, Q ∈ C ∞ 0 (C n ) with ord(Q) ≥ m + 1 and R * 1 , R * 2 have the same properties as those of R 1 , R 2 in (3.1), respectively. It follows from the definition that the Bloom-Graham type of M at p equals m < ∞, which is a contradiction.
From Lemma 6.2, the Taylor series of F at the origin in C n can be expressed as 2 α∈Z n + Re(c α z α ) where c α ∈ C. When the hypersurface M and p ∈ M are fixed, the sequence {c α } α∈Z n + is determined by the chosen standard coordinates (z, w). For a given coordinate (z, w), we define the formal power series Hereafter in this section, we only consider the case when the hypersurface admits a good coordinate at p. That is to say, M can be expressed as in the model form where F is the same as in (3.1). LetΓ be the set of (germs of) nonconstant holomorphic curvesγ = (γ 1 , . . . ,γ n ) : (C, 0) → (C n , 0). Let ∞ k=1 a jk t k be the Taylor series ofγ j for j = 1, . . . , n. After these Taylor series are substituted into S(z 1 , . . . , z n ) with z j =γ j (t), a formal computation gives the formal power series, denoted by (S •γ)(t), in the following.
The relationship between F and S implies that The following theorem is a natural generalization of Theorem 5.6. Let Γ denote the set of nonconstant holomorphic mappings γ : (C, 0) → (C n+1 , p). Theorem 6.3. Let M be a real hypersurface in C n+1 (n ≥ 1) admitting a good coordinate at p as in (6.2). If the Bloom-Graham type of M at p is infinity, then the following three conditions are equivalent.
(i) There exists a γ ∈ Γ tangent to M at p to infinite order; (ii) There exists a good coordinate (z, w) for M at p on which (F •γ)(t) is flat at t = 0 for someγ ∈Γ; (iii) There exists a good coordinate (z, w) for M at p such that the formal power series (S •γ)(t) in (6.3) converges on an open neighborhood of t = 0 for somê γ ∈Γ, where S is as in (6.1).
Proof. First, let us show (ii) =⇒ (i). Let γ(t) = (γ(t), 0) ∈ Γ, whereγ is as in (ii), and let r be a defining function for M as in (6.2). Then we have (r • γ)(t) = (F •γ)(t) = O(t N ) for every N ∈ N, which implies (i). Second, let us show (iii) =⇒ (ii). Let I := {j :γ j ≡ 0}. Define the map T I : C n → C n by (w 1 , . . . , w n ) = T I (z 1 , . . . , z n ) where w j = z j if j ∈ I and w j = 0 otherwise. It follows from Abel's lemma (cf. [12]) that the convergence of (S •γ)(t) for some t = 0 implies that the power series (S • T I )(z) converges on an open neighborhood of z = 0, which means that S • T I can be regarded as a holomorphic function there. Letting w * = w −(S •T I )(z), we can express the hypersurface M on the good coordinate (z, w * ) by the equation: 2Re(w * ) + F * (z,z) = 0, where F * (z,z) = F (z,z) − 2Re((S • T I )(z)). By using the equality S • Third, let us show (i) =⇒ (iii). We may assume that a holomorphic curve in (i) can be expressed as γ(t) = (γ(t), −h(t)), whereγ ∈Γ and h ∈ O 0 (C) with h(0) = 0, on a good coordinates (z, w). Note that (i) is equivalent to the condition (r • γ)(t) = O(t N ) for every N ∈ N. Let ∞ j=1 a j t j be the Taylor series of h at t = 0, which converges on an open neighborhood of t = 0. Now, after the first finite sum of the Taylor series of h and F •γ in (6.4) are substituted into the equations (r for every N ∈ N. From the above equality, we can see that c j = a j for every j ∈ N. This means that (S •γ)(t) converges and (S •γ)(t) = h(t) on an open neighborhood of t = 0.
In [9], smooth pseudoconvex real hypersurfaces in C n+1 of Bloom-Graham infinite type admitting no curve γ ∞ in (1.2) are constructed. It follows from Theorem 6.3 that many such hypersurfaces can be easily constructed. Proof. We remark that (ii) =⇒ (iii) can be shown by using Abel's lemma (cf. the proof of (iii) =⇒ (ii) in Theorem 6.3). The other implications can be directly obtained from Theorem 6.3.
Remark 6.5. From (iii) in the above corollary, it is easy to construct smooth pseudoconvex hypersurfaces of Bloom-Graham infinite type satisfying the condition (i). One of simple examples of hypersurfaces is given by the equation Re(w) + f 1 (z 1 ,z 1 ) + · · · + f n (z n ,z n ) = 0, where f j (j = 1, . . . , n) are subharmonic functions constructed in [9] (see also Remark 5.8). The example constructed in [9] takes the same form, but this example needs some additional conditions for each f j .
Roughly speaking, when the flatness of hypersurfaces is stronger, it becomes easier to find the curve tangent to M to higher order. Thus, the following question seems to be more difficult: does there exist a smooth pseudoconvex real hypersurface in C n+1 (n ≥ 2) of the Bloom-Graham finite type that admits no γ ∈ Γ tangent to M to infinite order? The following theorem gives an affirmative answer. Theorem 6.6. Let n ≥ 2. There exists a smooth pseudoconvex real hypersurface M in C n+1 with ∆ 1 (M, p) = ∞ and ∆ reg 1 (M, p) < ∞ (in particular, M is of Bloom-Graham finite type at p) that admits no γ ∈ Γ tangent to M at p to infinite order.
Proof. Notice that since a desired real hypersurface M satisfies that ∆ reg 1 (M, p) < ∆ 1 (M, p), there is no N -canonical coordinate for M from Theorem 2.3.
In particular, every γ ∈ Γ reg does not take the form (6.6), so it is easy to see ∆ reg 1 (M, 0) = 6. More precisely, we consider the case when γ takes the form (6.6). Lemma 6.7. Let N be an arbitrary integer with N ≥ 10. Then the following two conditions are equivalent.
Since R N +1 (t 2 , t 2 ) = O(t 2N +2 ) and the mixed terms and the pure terms cannot be canceled, g and h must satisfy that g(t) = O(t N +1 ) and h(t) + N j=8 c j t 2j = O(t 2N +2 ), which imply the condition (ii).
It follows from the above lemma that ∆ 1 (M, p) = ∞. Now, let us assume that there exists a curve γ ∞ ∈ Γ such that ord(r • γ ∞ ) ≥ N for every N ∈ N. Let γ ∞ (t) =: (γ 1 (t), γ 2 (t), γ 3 (t)). Since γ ∞ satisfies the condition (i) in Lemma 6.7, the condition (ii) implies that γ 3 (t) = − N j=8 c j t 2j + O(t 2N +2 ) for every N ∈ N. But, ∞ j=8 c j t 2j does not converge away from the origin, which is a contradiction to the holomorphy of γ 3 . As a result, we see that there exists no γ ∈ Γ tangent to M at the origin to infinite order.
In higher dimensional case C n+1 with n ≥ 3, the following F is considered: where f is the same as that in (6.5). It is easy to construct higher dimensional hypersurfaces satisfying the properties in the theorem by using F in (6.5) in a similar fashion to the three-dimensional case.

Open problems
Theorems 5.5 and 6.3 only treat real hypersurfaces of the model form as in (3.2). The following problem is naturally raised.