aa r X i v : . [ m a t h . C V ] A p r ON THE STEINNESS INDEX
JIHUN YUM
Abstract.
We introduce the concept of Steinness index related to the Stein neigh-borhood basis. We then show several results: (1) The existence of Steinness index isequivalent to that of strong Stein neighborhood basis. (2) On the Diederich-Fornæssworm domains in particular, we present an explicit formula relating the Steinnessindex to the well-known Diederich-Fornæss index. (3) The Steinness index is 1 ifa smoothly bounded pseudoconvex domain admits finitely many boundary points ofinfinite type. Introduction
Let Ω ⊂ C n ( n ≥
2) be a bounded domain with smooth boundary. A smooth function ρ defined on a neighborhood V of Ω is called a (global) defining function of Ω if Ω = { z ∈ V : ρ ( z ) < } and dρ ( z ) = 0 for all z ∈ ∂ Ω.1.1.
Diederich-Fornæss index.
The
Diederich-Fornæss exponent of ρ is defined by η ρ := sup { η ∈ (0 ,
1) : − ( − ρ ) η is strictly plurisubharmonic on Ω } . If there is no such η , then we define η ρ = 0. The Diederich-Fornæss index of
Ω isdefined by DF (Ω) := sup η ρ , where the supremum is taken over all defining functions ρ . We say that the Diederich-Fornæss index of Ω exists if DF (Ω) ∈ (0 , DF (Ω) exists, then there exists abounded strictly plurisubharmonic exhaustion function on Ω. In other words, Ω be-comes a hyperconvex domain. In 1977, Diederich and Fornæss ([3]) proved that C -smoothness of ∂ Ω implies the existence of DF (Ω).1.2. Steinness index.
We introduce the following definition. The
Steinness exponentof ρ is defined by e η ρ := inf { e η > ρ e η is strictly plurisubharmonic on Ω ∁ ∩ U for some neighborhood U of ∂ Ω } , where Ω ∁ := C n \ Ω. If there is no such e η , then we define e η ρ = ∞ . The Steinness indexof
Ω is defined by S (Ω) := inf e η ρ , This research was supported by the SRC-GAIA (NRF-2011-0030044) through the National ResearchFoundation of Korea (NRF) funded by the Ministry of Education. where the infimum is taken over all defining functions ρ . We say that the Steinnessindex of Ω exists if S (Ω) ∈ [1 , ∞ ). Ω is said to have a Stein neighborhood basis if for anyneighborhood V of Ω, there exists a pseudoconvex domain V such that Ω ⊂ V ⊂ V .If S (Ω) exists, then there exist a defining function ρ and η ∈ (1 , ∞ ) such that ρ η is strictly plurisubharmonic on Ω ∁ ∩ U . Thus Ω has a Stein neighborhood basis. Incontrast with Diederich-Fornæss index, the smoothness of boundary does not impliesthe existence of S (Ω); Diederich-Fornæss worm domains provide an example ([4]). Insection 3, we characterize the Steinness index by means of a differential inequality onthe set of all weakly pseudoconvex boundary points (Theorem 3.1). This theorem playsa crucial role in this paper.1.3. Strong Stein neighborhood basis. Ω ⊂⊂ C n is said to have a strong Steinneighborhood basis if there exist a defining function ρ of Ω and ǫ > ǫ := { z ∈ C n : ρ ( z ) < ǫ } is pseudoconvex for all 0 ≤ ǫ < ǫ . This implies the existence of a Stein neighborhoodbasis. In section 4, we show that the existence of S (Ω) is actually equivalent to theexistence of a strong Stein neighborhood basis (Theorem 4.1).1.4. Worm domains.
In 1977, Diederich and Fornæss ([4]) constructed bounded smoothdomains in C whose Diederich-Fornæss indices are strictly less than one. These exam-ples are called worm domains and the only known domains in C n which have non-trivialDiederich-Fornæss indices. Therefore, worm domains are worth to be studied.Recently, Liu calculated the exact value of the Diederich-Fornæss index of wormdomains (Definition 5.1) in 2017. In section 5, exploiting the idea of [10], we obtain acalculation of the exact values of the Steinness index of worm domains (Theorem 5.3).More precisely, the result is as follows. Theorem 1.1. If Ω β ( β > π ) is a worm domain, then the following 4 conditions areequivalent: < DF (Ω β ) < .
2. Ω β admits the Steinness index.
3. Ω β admits the Stein neighborhood basis.
4. Ω β admits a strong Stein neighborhood basis.Moreover, if one of the above conditions holds, then DF (Ω β ) + 1 S (Ω β ) = 2 . Sufficient conditions for DF ( Ω ) = ( Ω ) = If a bounded domain Ω ⊂ C n is strongly pseudoconvex, then there exists a strictly plurisubharmonic definingfunction of Ω, which implies DF (Ω) = 1 and S (Ω) = 1.In fact, more is known: If a smoothly bounded pseudoconvex domain Ω is B-regular(i.e., for every p ∈ ∂ Ω, there exists a continuous peak function at p ), then DF (Ω) = 1and S (Ω) = 1 ([12], [13]). Since finite type domains in the sense of D’Angelo are B-regular by Catlin ([2]), both indices are equal to 1. In section 6, we give an alternative N THE STEINNESS INDEX 3 proof of this fact using modified Theorem 3.1 (Corollary 6.4). Moreover, if the set ofall infinite type boundary points is finite, then the Steinness index is 1 (Corollary 6.5).In section 7, we also demonstrate that if Ω ⊂⊂ C n is a C -smooth convex domain, then DF (Ω) = 1 and S (Ω) = 1 (Corollary 7.2). For further results, we refer the reader to[5], [6], [7], [8] and [9]. 2. Preliminary
We first fix the notation of this paper, unless otherwise mentioned. • Ω : a bounded pseudoconvex domain with smooth boundary in C n . • ρ : a defining function of Ω. • Σ : the set of all weakly pseudoconvex points in ∂ Ω. • Σ ∞ : the set of all infinite type points in ∂ Ω. • g : the standard Euclidean complex Hermitian metric in C n . • ∇ : the Levi-Civita connection of g . • ∇ ρ : the real gradient of ρ . • U : a tubular neighborhood of ∂ Ω. • L ρ : the Levi-form of ρ .Define N ρ := 1 qP nj =1 | ∂ρ∂z j | n X j =1 ∂ρ∂ ¯ z j ∂∂z j . Let J be the complex structure of C n and T p ( ∂ Ω) be the real tangent space of ∂ Ω at p ∈ ∂ Ω. Let T cp ( ∂ Ω) := J ( T p ( ∂ Ω)) ∩ T p ( ∂ Ω). Then the complexified tangent space of T cp ( ∂ Ω), C T cp ( ∂ Ω) := C ⊗ T cp ( ∂ Ω), can be decomposed into the holomorphic tangentspace T , p ( ∂ Ω) and the anti-holomorphic tangent space T , p ( ∂ Ω). We call X a (1 , X ∈ T , p ( ∂ Ω).Let
X, Y, Z be complex vector fields in C n . A direct calculation implies the followingproperties. g ( N ρ , N ρ ) = 12 , N ρ ρ = vuut n X j =1 (cid:12)(cid:12)(cid:12)(cid:12) ∂ρ∂z j (cid:12)(cid:12)(cid:12)(cid:12) = k∇ ρ k ,N ρ + N ρ = 2 Re( N ρ ) = ∇ ρ k∇ ρ k , L ρ ( X, Y ) = g ( ∇ X ∇ ρ, Y ) = X ( Y ρ ) − ( ∇ X Y ) ρ,Zg ( X, Y ) = g ( ∇ Z X, Y ) + g ( X, ∇ Z Y ) , Zρ = g ( ∇ ρ, Z ) , ∇ X ∇ Y − ∇ Y ∇ X − ∇ [ X,Y ] = 0 . For p ∈ ∂ Ω whenever we mention lim z → p , it means z approaches p along the real normaldirection, and “smooth” means C ∞ -smooth, although C -smoothness suffices for ourpurpose.Now, we introduce two lemmas which we will use in the next section. JIHUN YUM
Lemma 2.1 ([10]) . Let Ω ⊂⊂ C n be a pseudoconvex domain with smooth boundary,and ρ be a defining function of Ω . Suppose that L is a (1 , tangent vector field so that L ρ ( L, L ) = 0 at p ∈ ∂ Ω . Assume that T j (1 ≤ j ≤ n − are (1 , tangent vector fieldsand T , T , · · · , T n − , L are orthogonal at p . Then L ρ ( L, T j ) = 0 for ≤ j ≤ n − at p . Lemma 2.2.
Let Ω ⊂⊂ C n be a pseudoconvex domain with smooth boundary, and ρ bea defining function of Ω . Let ψ be a smooth function defined on U and e ρ = ρe ψ . Wedenote e N = N e ρ , N = N ρ . Let e L , L be (1 , tangent vector fields on U such that e L = L on ∂ Ω , and e L e ρ = 0 , Lρ = 0 on U . Define Σ L := { p ∈ ∂ Ω : L ρ ( L, L )( p ) = 0 } . Then L e ρ ( e L, e N ) k∇ e ρ k = L ρ ( L, N ) k∇ ρ k + 12 ( Lψ ) and (2.1) e N L e ρ ( e L, e L ) k∇ e ρ k = N L ρ ( L, L ) k∇ ρ k + 12 L ψ ( L, L ) − | Lψ | − Re (cid:20) L ρ ( L, N ) k∇ ρ k ( Lψ ) (cid:21) on Σ L . In particular, | L e ρ ( e L, e N ) | k∇ e ρ k + 12 e N L e ρ ( e L, e L ) k∇ e ρ k = | L ρ ( L, N ) | k∇ ρ k + 12 N L ρ ( L, L ) k∇ ρ k + 14 L ψ ( L, L ) on Σ L .Proof. Since e L = L , e N = N , and k∇ e ρ k = e ψ k∇ ρ k on ∂ Ω, L e ρ ( e L, e N ) = L e ρ ( L, N )= e ψ L ρ ( L, N ) + ρ L e ψ ( L, N ) + ( Lρ )( N e ψ ) + ( Le ψ )( N ρ )= e ψ L ρ ( L, N ) + e ψ ( Lψ ) k∇ ρ k . Therefore, we have L e ρ ( e L, e N ) k∇ e ρ k = L ρ ( L, N ) k∇ ρ k + 12 ( Lψ )on Σ L . Now, we prove (2.1) holds at p ∈ Σ L . e N L e ρ ( e L, e L ) = N L e ρ ( e L, e L ) = N g ( ∇ e L ∇ e ρ, e L ) = g ( ∇ N ∇ e L ∇ e ρ, e L ) + g ( ∇ e L ∇ e ρ, ∇ N e L )(2.2) = g ( ∇ e L ∇ N ∇ e ρ, e L ) + g ( ∇ [ N, e L ] ∇ e ρ, e L ) + g ( ∇ e L ∇ e ρ, ∇ N e L ) . If L vanishes at p , then the last three terms of (2.2) are all zero and so e N L e ρ ( e L, e L ) = N L ρ ( L, L ) = 0. Thus (2.1) holds at p . Therefore, we assume that L = 0 at p andnormalize L and e L as g ( L, L ) = g ( e L, e L ) = . This normalization is for convenience and N THE STEINNESS INDEX 5 does not affect the result. Now, we will compute the last three terms of (2.2) one byone. Let {√ e T , · · · , √ e T n − , √ e L } be an orthonormal basis of T , p ( ∂ Ω). Then first, g ( ∇ e L ∇ e ρ, ∇ N e L )= g ∇ e L ∇ e ρ, n − X j =1 g ( ∇ N e L, √ e T j ) √ e T j + g ( ∇ N e L, √ e L ) √ e L + g ( ∇ N e L, √ N ) √ N ! = n − X j =1 g ( ∇ N e L, e T j ) L e ρ ( e L, e T j ) + 2 g ( ∇ N e L, e L ) L e ρ ( e L, e L ) + 2 g ( ∇ N e L, N ) L e ρ ( e L, N )=2 g ( ∇ N e L, N ) L e ρ ( e L, N )=2 g ( ∇ N e L, N ) L e ρ ( L, N ) . Here, we used Lemma 2.1 in the third equality above. Second, by the same argumentas above, g ( ∇ [ N, e L ] ∇ e ρ, e L ) = g ( ∇ e L ∇ e ρ, [ N, e L ]) = 2 g ([ N, e L ] , N ) L e ρ ( e L, N ) . Third, g ( ∇ e L ∇ N ∇ e ρ, e L )= g ( ∇ L ∇ N ∇ e ρ, L )= g ( ∇ N ∇ L ∇ e ρ, L ) − g ( ∇ [ N,L ] ∇ e ρ, L )= N g ( ∇ L ∇ e ρ, L ) − g ( ∇ L ∇ e ρ, ∇ N L ) − g ( ∇ [ N,L ] ∇ e ρ, L )= N L e ρ ( L, L ) − g ( ∇ N L, N ) L e ρ ( L, N ) − g ([ N, L ] , N ) L e ρ ( L, N ) . Therefore, we have e N L e ρ ( e L, e L ) = N L e ρ ( L, L )+ 2 (cid:16) g ( ∇ N e L, N ) − g ( ∇ N L, N ) (cid:17) L e ρ ( L, N )+ 2 (cid:16) g ([ N, e L ] , N ) − g ([ N, L ] , N ) (cid:17) L e ρ ( L, N ) . Now, L e ρ ( L, L ) = e ψ L ρ ( L, L ) + ρ L e ψ ( L, L ) + ( Lρ )( Le ψ ) + ( Le ψ )( Lρ ) ,N L e ρ ( L, L ) = e ψ (cid:0) N L ρ ( L, L ) + (
N ρ ) L ψ ( L, L ) + (
N ρ ) | Lψ | (cid:1) , (2.3)and L ρ ( N, L ) = N ( Lρ ) − ( ∇ N L ) ρ = − ( ∇ N L ) ρ ⇒ − L ρ ( L, N ) = ( ∇ N L ) ρ = 2 g ( ∇ N L, N )( N ρ ) = g ( ∇ N L, N ) k∇ ρ k⇒ g ( ∇ N L, N ) = − L ρ ( L, N ) k∇ ρ k . JIHUN YUM
By the same argument, g ( ∇ N e L, N ) = − L e ρ ( e L, N ) k∇ e ρ k = − L ρ ( L, N ) k∇ ρ k −
12 ( Lψ ) ⇒ g ( ∇ N e L, N ) − g ( ∇ N L, N ) = −
12 ( Lψ ) , (2.4)and g ([ N, L ] , N ) = g ( ∇ N L, N ) − g ( ∇ L N, N ) = g ( ∇ N L, N ) + g ( ∇ N − N L, N ) − g ( ∇ L N, N ) ,g ([ N, e L ] , N ) = g ( ∇ N e L, N ) − g ( ∇ e L N, N ) = g ( ∇ N e L, N ) + g ( ∇ N − N e L, N ) − g ( ∇ e L N, N ) . Here, g ( ∇ L N, N ) = g ( ∇ e L N, N ) on Σ L and since N − N is a real tangent vector fieldon ∂ Ω, g ( ∇ N − N L, N ) = g ( ∇ N − N e L, N ) on Σ L . Therefore, g ([ N, e L ] , N ) − g ([ N, L ] , N ) = g ( ∇ N e L, N ) − g ( ∇ N L, N ) = −
12 ( Lψ ) . (2.5)Combining (2.3), (2.4), and (2.5), we have e N L e ρ ( e L, e L )= e ψ (cid:0) N L ρ ( L, L ) + (
N ρ ) L ψ ( L, L ) + (
N ρ ) | Lψ | (cid:1) − ( Lψ ) (cid:0) e ψ L ρ ( L, N ) + e ψ ( Lψ )( N ρ ) (cid:1) − ( Lψ ) (cid:16) e ψ L ρ ( L, N ) + e ψ ( Lψ )( N ρ ) (cid:17) = e ψ (cid:2) N L ρ ( L, L ) + (
N ρ ) L ψ ( L, L ) − ( N ρ ) | Lψ | − (cid:0) L ρ ( L, N )( Lψ ) (cid:1)(cid:3) . Since k∇ e ρ k = e ψ k∇ ρ k on ∂ Ω, we conclude e N L e ρ ( e L, e L ) k∇ e ρ k = N L ρ ( L, L ) k∇ ρ k + 12 L ψ ( L, L ) − | Lψ | − (cid:20) L ρ ( L, N ) k∇ ρ k ( Lψ ) (cid:21) at p ∈ Σ L . (cid:3) Equivalent Definition for Steinness index
In order to check whether S (Ω) exists, by its definition, it is necessary to find adefining function ρ and η > ρ η is strictly plurisubharmonic on Ω ∁ ∩ U . Inthis section, we replace this condition on Ω ∁ ∩ U to another condition on Σ, so that weonly need to check it at weakly pseudoconvex boundary points. Now, we introduce themain theorem of this section. Theorem 3.1.
Let Ω ⊂⊂ C n be a pseudoconvex domain with smooth boundary, and ρ be a defining function of Ω . Let L be an arbitrary (1 , tangent vector field on ∂ Ω .Define Σ L := { p ∈ ∂ Ω : L ρ ( L, L )( p ) = 0 } . N THE STEINNESS INDEX 7
Let η ρ be the infimum of η ∈ (1 , ∞ ) satisfying η − | L ρ ( L, N ) | k∇ ρ k − N L ρ ( L, L ) k∇ ρ k ≤ on Σ L for all L . Here, N = N ρ and we extend L so that Lρ = 0 on U . Then S (Ω) = inf η ρ where the infimum is taken over all smooth defining functions ρ . The following sequence of lemmas are essential towards the proof of the theoremabove.
Lemma 3.2.
Let Ω ⊂⊂ C n be a pseudoconvex domain with smooth boundary, and ρ be a defining function of Ω . For p ∈ ∂ Ω , let U p be a neighborhood of p in C n . Let L be a smooth (1 , tangent vector field in U p such that Lρ = 0 and L ρ ( L, L ) = 0 . Fix η ∈ (1 , ∞ ) . We denote N ρ by N . Then L ρ η ( aL + bN, aL + bN ) > for all ( a, b ) ∈ C \ (0 , on Ω ∁ ∩ U p implies η − | L ρ ( L, N ) | k∇ ρ k − N L ρ ( L, L ) k∇ ρ k ≤ at p .Proof. By using Lρ = 0 and the assumption, we have L ρ η ( aL + bN, aL + bN )= η ρ η − (cid:20) | a | L ρ ( L, L ) + 2Re (cid:0) ab L ρ ( L, N ) (cid:1) + | b | (cid:18) L ρ ( N, N ) + η − ρ | N ρ | (cid:19)(cid:21) > a, b ) ∈ C \ (0 ,
0) on Ω ∁ ∩ U p , and it is equivalent to(3.1) | a | L ρ ( L, L ) − | a || b || L ρ ( L, N ) | + | b | (cid:18) L ρ ( N, N ) + η − ρ | N ρ | (cid:19) > a, b ) ∈ C \ (0 ,
0) on Ω ∁ ∩ U p . This is because η ρ η − > a or b , one can make 2Re (cid:0) ab L ρ ( L, N ) (cid:1) = − | a || b || L ρ ( L, N ) | . We may assume that L ρ ( N, N )+ η − ρ | N ρ | > ∁ ∩ U p , because it blows up as point goes to the boundary.By divding (3.1) by | a | and letting x = | b || a | , (3.1) is equivalent to(3.2) L ρ ( L, L ) − x | L ρ ( L, N ) | + x (cid:18) L ρ ( N, N ) + η − ρ | N ρ | (cid:19) > x ≥ ∁ ∩ U p . The axis of symmetry of the quadratic equation above is | L ρ ( L, N ) | L ρ ( N, N ) + η − ρ | N ρ | , JIHUN YUM which is always positive on Ω ∁ ∩ U p . Thus, (3.2) if and only if the determinent(3.3) | L ρ ( L, N ) | − ( L ρ ( L, L )) (cid:18) L ρ ( N, N ) + η − ρ | N ρ | (cid:19) < ∁ ∩ U p . Now, taking a limit to p on (3.3) along the real normal direction yields(3.4) | L ρ ( L, N ) | − ( η − | N ρ | lim z → p L ρ ( L, L ) ρ ≤ p ∈ ∂ Ω. Here,(3.5) lim z → p L ρ ( L, L )( z ) ρ ( z ) = − ( N + N ) L ρ ( L, L )( p ) − ( N + N ) ρ ( p ) = N L ρ ( L, L )( p ) N ρ ( p ) . The explanation of second equality of (3.5) is following. Since L ρ ( L, L ) = 0 at p ∈ ∂ Ω, L ρ ( L, L ) attains the local minimum at p on ∂ Ω. Thus, the directional derivative of L ρ ( L, L ) along the real tangent vector N − N at p is zero, and ( N − N ) L ρ ( L, L ) = 0implies N L ρ ( L, L ) = N L ρ ( L, L ) at p . Finally, since N ρ = k∇ ρ k , (3.4) is equivalent to1 η − | L ρ ( L, N ) | k∇ ρ k − N L ρ ( L, L ) k∇ ρ k ≤ p ∈ ∂ Ω. (cid:3) Lemma 3.3.
Let Ω ⊂⊂ C n be a pseudoconvex domain with smooth boundary, and ρ bea defining function of Ω . Let { U α } be a chart of U and L be a non-vanishing smooth (1 , tangent vector field in U α such that Lρ = 0 . We normalize L as g ( L, L ) = . Fix η ∈ (1 , ∞ ) . We denote N ρ by N . Define Σ αL := { p ∈ ∂ Ω ∩ U α : L ρ ( L, L )( p ) = 0 } . If (3.6) 1 η − | L ρ ( L, N ) | k∇ ρ k − N L ρ ( L, L ) k∇ ρ k < on Σ αL , then there exists a neighborhood V αL of Σ αL in U α such that L ρ η ( aL + bN, aL + bN ) > for all ( a, b ) ∈ C \ (0 , on Ω ∁ ∩ V αL .Proof. First, note that the assumption (3.6) is equivalent to | L ρ ( L, N ) | − ( η − N ρ )( N L ρ ( L, L )) < αL . Now define F : Ω ∁ ∩ U α → R by F ( z ) := | L ρ ( L, N )( z ) | − ( L ρ ( L, L )( z ) − L ρ ( L, L )( p )) L ρ ( N, N )( z ) − ( η − L ρ ( L, L )( z ) − L ρ ( L, L )( p ) ρ ( z ) | N ρ ( z ) | N THE STEINNESS INDEX 9 for all z ∈ Ω ∁ ∩ U α , where p ∈ ∂ Ω ∩ U α is the closest point to z , and F ( z ) := | L ρ ( L, N )( z ) | − ( η − N ρ ( z ))( N L ρ ( L, L )( z ))for all z ∈ ∂ Ω ∩ U α . Since lim z → p F ( z ) = F ( p ) for all p ∈ ∂ Ω ∩ U α , F is continuous onΩ ∁ ∩ U α . By the assumption (3.6), there exists a neighborhood V αL of Σ αL in U α suchthat | L ρ ( L, N )( z ) | − ( L ρ ( L, L )( z ) − L ρ ( L, L )( p )) L ρ ( N, N )( z ) − ( η − L ρ ( L, L )( z ) − L ρ ( L, L )( p ) ρ ( z ) | N ρ ( z ) | = | L ρ ( L, N )( z ) | − ( L ρ ( L, L )( z ) − L ρ ( L, L )( p )) (cid:18) L ρ ( N, N )( z ) + η − ρ ( z ) | N ρ ( z ) | (cid:19) < z ∈ Ω ∁ ∩ V αL . We may assume that L ρ ( N, N )( z ) + η − ρ ( z ) | N ρ ( z ) | > z ∈ Ω ∁ ∩ V αL , because it blows up as z goes to the boundary. Therefore, the followingquadratic function ( L ρ ( L, L )( z ) − L ρ ( L, L )( p )) − x | L ρ ( L, N )( z ) | + x (cid:18) L ρ ( N, N )( z ) + η − ρ ( z ) | N ρ ( z ) | (cid:19) > z ∈ Ω ∁ ∩ V αL , x ≥
0. By letting x = | b || a | and multiplying | a | , we have | a | ( L ρ ( L, L )( z ) − L ρ ( L, L )( p )) − | a || b | | L ρ ( L, N )( z ) | (3.7) + | b | (cid:18) L ρ ( N, N )( z ) + η − ρ ( z ) | N ρ ( z ) | (cid:19) > z ∈ Ω ∁ ∩ V αL , ( a, b ) ∈ C \ (0 , | a | = 0, then (3.7) is equivalent to L ρ ( N, N )( z ) + η − ρ ( z ) | N ρ ( z ) | >
0, which is automatically satisfied. Now, (3.7) implies | a | ( L ρ ( L, L )( z ) − L ρ ( L, L )( p )) + 2 Re (cid:0) ab ( L ρ ( L, N )( z )) (cid:1) + | b | (cid:18) L ρ ( N, N )( z ) + η − ρ ( z ) | N ρ ( z ) | (cid:19) > z ∈ Ω ∁ ∩ V αL , ( a, b ) ∈ C \ (0 ,
0) because Re (cid:0) ab ( L ρ ( L, N )) (cid:1) > −| a || b | | L ρ ( L, N ) | .Since Ω is pseudoconvex, L ρ ( L, L )( p ) ≥ p ∈ ∂ Ω, and thus | a | ( L ρ ( L, L )( z )) + 2 Re (cid:0) ab ( L ρ ( L, N )( z )) (cid:1) (3.8) + | b | (cid:18) L ρ ( N, N )( z ) + η − ρ ( z ) | N ρ ( z ) | (cid:19) > z ∈ Ω ∁ ∩ V αL , ( a, b ) ∈ C \ (0 , Lρ = 0, (3.8) is equivalent to L ρ η ( aL + bN, aL + bN )( z ) > for all z ∈ Ω ∁ ∩ V αL , ( a, b ) ∈ C \ (0 , (cid:3) Lemma 3.4.
Let Ω ⊂⊂ C n be a pseudoconvex domain with smooth boundary, and ρ bea defining function of Ω . Let { U α } be a chart of U and L be a non-vanishing smooth (1 , tangent vector field in U α such that Lρ = 0 . We normalize L as g ( L, L ) = . Fix η ∈ (1 , ∞ ) . We denote N ρ by N . Define Σ αL := { p ∈ ∂ Ω ∩ U α : L ρ ( L, L )( p ) = 0 } . If η − | L ρ ( L, N ) | k∇ ρ k − N L ρ ( L, L ) k∇ ρ k ≤ on Σ αL , then there exists a defining function e ρ such that for any small ν > so that η − − ν > , e η − | L e ρ ( e L, e N ) | k∇ e ρ k − e N L e ρ ( e L, e L ) k∇ e ρ k < on Σ αL , where e η = 1 + η − − ν and e N = N e ρ . e L is a non-vanishing smooth (1 , tangentvector field in U α such that e L = L on ∂ Ω ∩ U α , and e L e ρ = 0 on U α .Proof. First, notice that e η − = η − − ν . Define e ρ = ρe ǫψ , where ψ = k z k , ǫ is a smallpositive number and we will decide ǫ later. Then by Lemma 2.2,1 e η − | L e ρ ( e L, e N ) | k∇ e ρ k − e N L e ρ ( e L, e L ) k∇ e ρ k = (cid:18) η − − ν (cid:19) | L ρ ( L, N ) | k∇ ρ k − N L ρ ( L, L ) k∇ ρ k + (cid:18) η − − ν (cid:19) (cid:18) ǫ | Lψ | + ǫ Re (cid:20) L ρ ( L, N ) k∇ ρ k ( Lψ ) (cid:21)(cid:19) − ǫ ≤ (cid:18) η − − ν (cid:19) | L ρ ( L, N ) | k∇ ρ k − N L ρ ( L, L ) k∇ ρ k + (cid:18) η − − ν (cid:19) ǫ | Lψ | + ν | L ρ ( L, N ) | k∇ ρ k + (cid:16) η − + 1 − ν (cid:17) ν | Lψ | ǫ − ǫ ≤ η − | L ρ ( L, N ) | k∇ ρ k − N L ρ ( L, L ) k∇ ρ k− ν | L ρ ( L, N ) | k∇ ρ k + (cid:18) η − − ν (cid:19)
14 + η − + 1 − ν ν ! | Lψ | ǫ − ǫ αL . Now η − | L ρ ( L,N ) | k∇ ρ k − N L ρ ( L,L ) k∇ ρ k ≤ − ν | L ρ ( L,N ) | k∇ ρ k ≤
0. Wechoose sufficiently small ǫ > (cid:16) η − + 1 − ν (cid:17) (cid:18) + η − +1 − ν ν (cid:19) | Lψ | ǫ − ǫ < N THE STEINNESS INDEX 11 e η − | L e ρ ( e L, e N ) | k∇ e ρ k − e N L e ρ ( e L, e L ) k∇ e ρ k < αL . (cid:3) Lemma 3.5.
Let Ω ⊂⊂ C n be a pseudoconvex domain with smooth boundary, and Π be the set of all strongly pseudoconvex points in ∂ Ω . Then for any defining function ρ of Ω and η ∈ (1 , ∞ ) , there exists a neighborhood V of Π in C n such that ρ η is strictlyplurisubharmonic on Ω ∁ ∩ V .Proof. Let V be a neighborhood of Π in C n , and { V α } be a chart of V . We denote N = N ρ . It is enough to show that(3.9) L ρ η ( aL + bN, aL + bN ) > a, b ) ∈ C \ (0 ,
0) on Ω ∁ ∩ V α , where L is an arbitrary non-vanishing smooth(1 ,
0) tangent vector field in V α . As in the proof of Lemma 3.2, (3.9) is equivalent to(3.10) | L ρ ( L, N ) | − ( L ρ ( L, L )) (cid:18) L ρ ( N, N ) + η − ρ | N ρ | (cid:19) < ∁ ∩ V α . After possibly shrinking V , (3.10) holds because L ρ ( L, L ) > L ρ ( N, N ) + η − ρ | N ρ | blows up as point approaches to ∂ Ω. (cid:3) Proof of Theorem 3.1.
First, we prove S (Ω) ≥ inf η ρ . For a defining function ρ and η ∈ (1 , ∞ ), if ρ η is strictly plurisubharmonic on Ω ∁ ∩ U , then L ρ η ( aL + bN, aL + bN ) > a, b ) ∈ C \ (0 ,
0) on Ω ∁ ∩ U . By Lemma 3.2, this implies that1 η − | L ρ ( L, N ) | k∇ ρ k − N L ρ ( L, L ) k∇ ρ k ≤ L for all L . Therefore, S (Ω) ≥ inf η ρ .Next, we prove S (Ω) ≤ inf η ρ . By Lemma 3.5, we only need to consider it in aneighborhood of Σ in C n . Let η = inf η ρ . Fix e η > η and choose small ν > η := 1 + 1 e η − + ν ⇔ e η = 1 + 1 η − − ν ! satisfies e η > η > η . Let { U α } be a chart of U and L be a non-vanishing smooth (1 , ∂ Ω ∩ U α such that g ( L, L ) = . Since η > η , there exists ρ such that 1 η − | L ρ ( L, N ) | k∇ ρ k − N L ρ ( L, L ) k∇ ρ k ≤ on Σ αL for all L . Here, we extend L so that Lρ = 0 on U α . By Lemma 3.4, there existsa defining function e ρ such that1 e η − | L e ρ ( e L, e N ) | k∇ e ρ k − e N L e ρ ( e L, e L ) k∇ e ρ k < αL for all L . Since e η is arbitrary, by Lemma 3.3, we have S (Ω) ≤ inf η ρ . (cid:3) Together with Lemma 2.2, we have
Corollary 3.6.
Let Ω ⊂⊂ C n be a pseudoconvex domain with smooth boundary, and ρ be a defining function of Ω . Let L be an arbitrary (1 , tangent vector field on ∂ Ω .Define Σ L := { p ∈ ∂ Ω : L ρ ( L, L )( p ) = 0 } . Let ψ be a smooth function defined on U , and η ψ be the infimum of η ∈ (1 , ∞ ) satisfying (cid:18) η − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) L ρ ( L, N ) k∇ ρ k + 12 ( Lψ ) (cid:12)(cid:12)(cid:12)(cid:12) − (cid:18) | L ρ ( L, N ) | k∇ ρ k + 12 N L ρ ( L, L ) k∇ ρ k (cid:19) − L ψ ( L, L ) ≤ on Σ L for all L . Here, N = N ρ and we extend L so that Lρ = 0 on U . Then S (Ω) = inf η ψ , where the infimum is taken over all smooth functions ψ . Strong Stein neighborhood basis
In 2012, Sahuto˘glu ([11]) gave several characterizations for Ω to have a strong Steinneighborhood basis. In this section, using one of the characterizations, we prove thatthe existence of the Steinness index and the existence of a strong Stein neighborhoodbasis are equivalent.
Theorem 4.1.
Let Ω ⊂⊂ C n be a pseudoconvex domain with smooth boundary. Then S (Ω) exists if and only if Ω has a strong Stein neighborhood basis.Proof. First, assume S (Ω) exists. Then there exists a defining function ρ of Ω and η ∈ (1 , ∞ ) such that ρ η is strictly plurisubharmonic on Ω ∁ ∩ U . Since level sets of ρ η and ρ are same, ρ is the desired defining function.Now, suppose that Ω has a strong Stein neighborhood basis. Then by Sahuto˘glu([11]), there exist a defining function ρ of Ω and c > L ρ ( L, L ) ≥ cρ k L k on Ω ∁ ∩ U , where L is a smooth (1 ,
0) tangent vector field in U with Lρ = 0. By letting k L k = and dividing it by ρ , we have L ρ ( L, L ) ρ ≥ c N THE STEINNESS INDEX 13
Define Σ L := { p ∈ Σ : L ρ ( L, L )( p ) = 0 } and denote N = N ρ . Taking a limit to p ∈ Σ L along the real normal direction gives12 N L ρ ( L, L ) k∇ ρ k ≥ c L . On the other hands, since Σ and { L ∈ T , p ( ∂ Ω) : k L k = } are compact, | L ρ ( L, N ) | k∇ ρ k has the maximum value M ≥ η ∈ (1 , ∞ )1 η − | L ρ ( L, N ) | k∇ ρ k − N L ρ ( L, L ) k∇ ρ k ≤ Mη − − c L . By choosing η > Mc + 1, we have1 η − | L ρ ( L, N ) | k∇ ρ k − N L ρ ( L, L ) k∇ ρ k ≤ L . By Theorem 3.1, we proved S (Ω) exists. (cid:3) Steinness index of worm domains
In this section, we calculate the exact value of the Steinness index of worm domains,and prove Theorem 1.1. Recall first the definition of worm domains.
Definition 5.1.
The worm domain Ω β ( β > π ) is defined byΩ β := (cid:26) ( z, w ) ∈ C : ρ ( z, w ) = (cid:12)(cid:12)(cid:12) z − e i log | w | (cid:12)(cid:12)(cid:12) − (1 − φ (log | w | )) < (cid:27) where φ : R → R is a fixed smooth function with the following properties :1. φ ( x ) ≥ φ is even and convex.2. φ − (0) = I β − π = [ − ( β − π ) , β − π ] . ∃ a > φ ( x ) > x < − a or x > a .4. φ ′ ( x ) = 0 if φ ( x ) = 1.Let ρ ( z, w ) = (cid:12)(cid:12)(cid:12) z − e i log | w | (cid:12)(cid:12)(cid:12) − (1 − φ (log | w | )), and U be a neighborhood of ∂ Ω β .Define L = 1 q(cid:12)(cid:12) ∂ρ∂z (cid:12)(cid:12) + (cid:12)(cid:12) ∂ρ∂w (cid:12)(cid:12) (cid:18) ∂ρ∂w ∂∂z − ∂ρ∂z ∂∂w (cid:19) ,N = 1 q(cid:12)(cid:12) ∂ρ∂z (cid:12)(cid:12) + (cid:12)(cid:12) ∂ρ∂w (cid:12)(cid:12) (cid:18) ∂ρ∂z ∂∂z + ∂ρ∂w ∂∂w (cid:19) . Let Σ be the set of all weakly pseudoconvex points in ∂ Ω β . ThenΣ = n (0 , w ) ∈ C : (cid:12)(cid:12) log | w | (cid:12)(cid:12) ≤ β − π o . By direct calculation, we have k∇ ρ k = 2 , L = e − i log | w | ∂∂w , N = − e i log | w | ∂∂z , L ρ ( L, N ) = iw e − i log | w | , N L ρ ( L, L ) = − | w | on Σ. Lemma 5.2 (Riccati equations [10]) . For a, b > and t > , the following Riccatieuqation ddt s ( t ) = a ( s ( t )) − s ( t ) t + bt has the solution s ( t ) = − r ba cot (cid:16) √ ab log t + φ (cid:17) t for arbitrary φ . Theorem 5.3.
Let Ω β ( β > π ) be a worm domain. Then S (Ω β ) = ( π π − β ) for π < β < π ∞ for π ≤ β Proof.
We will use Corollary 3.6. Let α = η − + 1 for η ∈ (1 , ∞ ). Suppose that thereexists a smooth function ψ defined on U such that(5.1) α (cid:12)(cid:12)(cid:12)(cid:12) L ρ ( L, N ) k∇ ρ k + 12 ( Lψ ) (cid:12)(cid:12)(cid:12)(cid:12) − (cid:18) | L ρ ( L, N ) | k∇ ρ k + 12 N L ρ ( L, L ) k∇ ρ k (cid:19) − L ψ ( L, L ) ≤ | L ρ ( L, N ) | k∇ ρ k + 12 N L ρ ( L, L ) k∇ ρ k = 14 | w | − | w | = 0on Σ. Consequently, (5.1) is equivalent to(5.2) α (cid:12)(cid:12)(cid:12)(cid:12) iw + ∂ψ∂w (cid:12)(cid:12)(cid:12)(cid:12) − ∂ ψ∂w∂w ≤ w = re iθ . Then (5.2) implies α (cid:18) ∂ψ∂w · ∂ψ∂w + 1 | w | + 2 Re (cid:18) ∂ψ∂w · iw (cid:19)(cid:19) − ∂ ψ∂w∂w = α ψ r + α r ψ θ + αr − αr ψ θ − ψ rr − r ψ r − r ψ θθ ≤ N THE STEINNESS INDEX 15 on Σ. Notice that R π ψ θ dθ = 0, R π ψ θθ dθ = 0 and R π ψ θ dθ ≥
0. Also by Schwarz’slemma (cid:16)R π ψ r dθ (cid:17) ≤ (cid:16)R π dθ (cid:17) (cid:16)R π ψ r dθ (cid:17) . Thus integrating on the both sides of (5.3)with respect to θ gives α π (cid:18)Z π ψ r dθ (cid:19) + 2 παr − Z π ψ rr dθ − r Z π ψ r dθ ≤ r ∈ [ e − ( β − π ) , e β − π ]. Define Ψ( r ) := π R π ψ ( r, θ ) dθ . Then (5.4) is equivalent to πα r + 2 παr − π rr − π r Ψ r ≤ r ∈ [ e − ( β − π ) , e β − π ], where Ψ r ( r ) = ddr Ψ( r ). Letting s ( r ) = Ψ r ( r ), we have − s ′ + αs − sr + 4 αr ≤ r ∈ [ e − ( β − π ) , e β − π ]. Suppose s (1) = s . By the comparison principle of ordinarydifferential equation, and Lemma 5.2 s ( r ) ≥ − α log r + φ ) r for all r ∈ [ e − ( β − π ) , e β − π ], where φ is a constant such that s = − φ ) . Since s ( r ) is a smooth function on [ e − ( β − π ) , e β − π ], the period of the cotangent functionis π , and − α (cid:0) β − π (cid:1) ≤ α log r ≤ α (cid:0) β − π (cid:1) , the following must hold. α (cid:16) β − π (cid:17) < π η − < π − β )2 β − π . (5.5)If β ≥ π , then (5.5) never holds. This proves that S (Ω β ) = ∞ for all β ≥ π .Now we assume π < β < π . Then (5.5) is equivalent to η > π π − β ) . (5.6)This shows that there does not exist any smooth function ψ defined on U satisfying(5.1) if η ≤ π π − β ) . Therefore, S (Ω β ) ≥ π π − β ) . Next, we prove that there exists asmooth function ψ defined on U satisfying (5.1) if η > π π − β ) . It is sufficient to find asmooth function ψ on Σ because we can extend ψ to U . By the argument above, (5.1)is equivalent to α ψ r + α r ψ θ + αr − αr ψ θ − ψ rr − r ψ r − r ψ θθ ≤ . (5.7) We assume ψ ( r, θ ) is independent of θ and let s ( r ) = ψ r . Then (5.7) becomes − s ′ + αs − sr + 4 αr ≤ . (5.8)By Lemma 5.2, − s ′ + αs − sr + αr = 0 has a solution s ( r ) = − (cid:0) α log r + π (cid:1) r . Since η > π π − β ) , s ( r ) is a well-defined smooth function on [ e − ( β − π ) , e β − π ] and hence ψ ( r, θ ) = R s ( r ) dr satisfies (5.1) on Σ. Therefore, by Corollary 3.6, S (Ω β ) = π π − β ) . (cid:3) Proof of Theorem 1.1.
Notice that the second and fourth conditions are equivalentby Theorem 4.1. We will show that the ranges of β for each condition are same. First,Liu ([10]) showed DF (Ω β ) = π β , which implies that < DF (Ω β ) < π < β < π . Next, by Theorem 5.3, the existence of S (Ω β ) is equivalent to π < β < π .Finally, one can prove that the third condition is equivalent to π < β < π usingTheorem 5.1 and 5.2 in [1] (see also section 5 in [1]). If one of the conditions holds,then DF (Ω β ) = π β and S (Ω β ) = π π − β ) implies the last equality. (cid:3) Steinness index of finite type domains
Theorem 3.1 says that the Steinness index is characterized by some differential in-equality on the set of all weakly pseudoconvex boundary points Σ. Here, we showthat considering the set Σ ∞ of infinite type boundary points suffices to characterize theSteinness index (Theorem 6.3). Lemma 6.1 ([11]) . Let Ω ⊂⊂ C n be a domain with smooth boundary, and K be acompact subset of ∂ Ω . Assume that z is of finite type for every z ∈ K and h ∈ C ∞ (Ω) is given. Then for every j > there exists h j ∈ C ∞ (Ω) such that | h j − h | ≤ j uniformlyon Ω and L h j ( X, X ) ≥ j k X k for all X ∈ T , p ( C n ) on K . Lemma 6.2.
Let Ω ⊂⊂ C n be a domain with smooth boundary, and Σ ∞ ⊂ ∂ Ω be the setof all infinite type boundary points. Assume that there exist a neighborhood V of Σ ∞ in C n , a defining function ρ of Ω , and η > such that ρ η is strictly plurisubharmonic on Ω ∁ ∩ V . Then there exists a defining function e ρ such that e ρ η is strictly plurisubharmonicon Ω ∁ ∩ U .Proof (Based on [11]) . Let Σ f be the set of all finite type boundary points, and Π bethe set of all strongly pseudoconvex boundary points. Let Σ := Σ f \ Π. If Σ = ∅ ,then by Lemma 3.5, ρ η is strictly plurisubharmonic on Ω ∁ ∩ U . Assume that Σ isnon-empty.Let χ : R → R be a smooth increasing convex function such that χ ( t ) = 0 if t ≤ χ ( t ) > t >
0. Let β = η − + 1 and A := max { βχ ′ ( t ) : 0 ≤ t ≤ } . Then N THE STEINNESS INDEX 17 by Lemma 6.1, there exist a sequence of φ j ∈ C ∞ (Ω) and neighborhoods V , V , V in C n with Σ ∞ ⊂⊂ V ⊂⊂ V ⊂⊂ V ⊂⊂ V such that :1. − A < φ j < − ln A on Ω.2. − A < φ j < − A on V ∩ ∂ Ω.3. − A < φ j < − ln A on ∂ Ω \ V .4. L φ j ( X, X ) > jA k X k for all X ∈ T , p ( C n ) on ∂ Ω \ V .Let h j = e φ j , a = A − , χ a ( t ) = χ ( t − a ), ψ j = χ a ◦ h j and e ρ j = ρe ψ j . Then ψ j ≡ U of V ∩ ∂ Ω in C n , hence e ρ j η is strictly plurisubharmonic on Ω ∁ ∩ U .Also, L h j ( X, X ) = e φ j L φ j ( X, X ) + e φ j | Xφ j | (6.1) = e φ j L φ j ( X, X ) + e − φ j | Xh j | > j k X k + A | Xh j | for all X ∈ T , p ( C n ) on ∂ Ω \ V .Let { U α } be a chart of U . Let e L, L be non-vanishing smooth (1 ,
0) tangent vectorfields in U α such that e L = L on ∂ Ω ∩ U α , and e L e ρ j = 0, Lρ = 0 on U α . We normalize e L, L as k e L k = k L k = . Denote e N = N e ρ , N = N ρ . DefineΣ αL := { p ∈ (Σ \ V ) ∩ U α : L ρ ( L, L )( p ) = 0 } . Let F ( ρ, η ) = η − | L ρ ( L,N ) | k∇ ρ k − N L ρ ( L,L ) k∇ ρ k . Then by using Lemma 2.2, on Σ αL , F ( e ρ j , η ) ≤ F ( ρ, η ) + β | L ρ ( L, N ) |k∇ ρ k | Lψ j | + β | Lψ j | − L ψ j ( L, L ) . (6.2)Here, β | L ρ ( L, N ) |k∇ ρ k | Lψ j | = βχ ′ a ( h j ) | L ρ ( L, N ) |k∇ ρ k | Lh j | ≤ χ ′ a ( h j ) (cid:18) β | L ρ ( L, N ) | k∇ ρ k + | Lh j | (cid:19) ,β | Lψ j | = β χ ′ a ( h j )) | Lh j | , − L ψ j ( L, L ) = − χ ′ a ( h j ) L h j ( L, L ) − χ ′′ a ( h j ) | Lh j | < − χ ′ a ( h j )8 j − χ ′ a ( h j )4 A | Lh j | − χ ′′ a ( h j )4 | Lh j | . All together, the right-hand side of (6.2) is negative if and only if(6.3) F ( ρ, η ) χ ′ a ( h j ) + β | L ρ ( L, N ) | k∇ ρ k + (cid:18) βχ ′ a ( h j )4 − A − χ ′′ a ( h j )4 χ ′ a ( h j ) (cid:19) | Lh j | − j < . Since h j − a < A − − A − < βχ ′ a ( h j ) ≤ A by the definition of A . Since ∂ Ω iscompact, β | L ρ ( L,N ) | k∇ ρ k is bounded. Since ρ η is strictly plurisubharmonic on V , F ( ρ, η ) ≤ ∂ Ω ∩ V by Lemma 3.2. For the outside of V , F ( ρ, η ) may be positive, but since χ ′ a ( h j ) > c on ∂ Ω \ V for some positive constant c > F ( ρ,η ) χ ′ a ( h j ) is bounded. Therefore,there exists a sufficiently large j > F ( e ρ j , η ) < αL for all α and e L . By Lemma 3.3, there exists a neighborhood U of Σ \ V suchthat e ρ j η is strictly plurisubharmonic on Ω ∁ ∩ U . Finally, by Lemma 3.5, there existsa neighborhood U of Π such that e ρ jη is strictly plurisubharmonic on Ω ∁ ∩ U . Since U ∪ U ∪ U is a neighborhood of ∂ Ω, the proof is completed. (cid:3)
Theorem 6.3.
Let Ω ⊂⊂ C n be a pseudoconvex domain with smooth boundary, and ρ be a defining function of Ω . Let L be an arbitrary (1 , tangent vector field on ∂ Ω , and Σ ∞ be the set of all infinite type boundary points. Define Σ ∞ ,L := { p ∈ Σ ∞ : L ρ ( L, L )( p ) = 0 } . Let η ρ be the infimum of η ∈ (1 , ∞ ) satisfying η − | L ρ ( L, N ) | k∇ ρ k − N L ρ ( L, L ) k∇ ρ k ≤ on Σ ∞ ,L for all L . Here, N = N ρ and we extend L so that Lρ = 0 on U . Then S (Ω) = inf η ρ where the infimum is taken over all smooth defining functions ρ .Proof. The proof is same as that of Theorem 3.1 except that we use Lemma 6.2 insteadof Lemma 3.5. (cid:3)
Corollary 6.4.
Let Ω ⊂⊂ C n be a pseudoconvex domain with smooth boundary. As-sume that all boundary points of Ω are of finite type. Then S (Ω) = 1 . Corollary 6.5.
Let Ω ⊂⊂ C n be a pseudoconvex domain with smooth boundary. As-sume that the set of all infinite type boundary points of Ω is finite. Then S (Ω) = 1 .Proof. Suppose that the set of all infinite type boundary points is one point, saysΣ ∞ = { p } . We denote e N = N e ρ , N = N ρ . Let e L , L be (1 ,
0) tangent vector fields on U such that e L = L on ∂ Ω, e L e ρ = 0, Lρ = 0 on U , and L ρ ( L, L ) = 0 at p . First, byLemma 2.2,(6.4) | L e ρ ( e L, e N ) | k∇ e ρ k + 12 e N L e ρ ( e L, e L ) k∇ e ρ k = | L ρ ( L, N ) | k∇ ρ k + 12 N L ρ ( L, L ) k∇ ρ k + 14 L φ ( L, L ) , where e ρ = ρe φ . By letting φ = α k z k and choosing α > p for all e L . Hence, we may assume thatthere exists a defining function ρ of Ω such that(6.5) | L ρ ( L, N ) | k∇ ρ k + 12 N L ρ ( L, L ) k∇ ρ k > L at p . Now, if L ( z ) = P nj =1 a j ( z ) ∂∂z j , then − L ρ ( L,N ) k∇ ρ k can be represented by − L ρ ( L, N ) k∇ ρ k ( z ) = n X j =1 b j ( z ) a j ( z ) N THE STEINNESS INDEX 19 for some complex-valued functions b j ( z ). Define ψ ( z ) = P nj =1 b j ( p ) z j . Then Lψ ( p ) = − L ρ ( L, N ) k∇ ρ k ( p ) , L ψ ( L, L )( p ) = 0 . If e ρ = ρe ψ , then by Lemma 2.2 and (6.5) at p η − | L e ρ ( e L, e N ) | k∇ e ρ k + 12 e N L e ρ ( e L, e L ) k∇ e ρ k (6.6)= (cid:18) η − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) L ρ ( L, N ) k∇ ρ k + 12 ( Lψ ) (cid:12)(cid:12)(cid:12)(cid:12) − (cid:18) | L ρ ( L, N ) | k∇ ρ k + 12 N L ρ ( L, L ) k∇ ρ k (cid:19) − L ψ ( L, L )= − (cid:18) | L ρ ( L, N ) | k∇ ρ k + 12 N L ρ ( L, L ) k∇ ρ k (cid:19) < L and η >
1. Therefore, by Theorem 6.3, S (Ω) = 1. If the number of points inΣ ∞ is more than 1, then one can construct a smooth function ψ on U satisfying (6.6)at all infinite type points using similar argument as above. (cid:3) Steinness index of convex domains
Assume that Ω ⊂⊂ C n is a domain with C k ( k ≥ DF (Ω) = 1 and S (Ω) = 1. In fact, we provemore: there exists a defining function which is strictly convex on C n \ ∂ Ω. For this, weneed the following notion of smooth maximum . For ǫ >
0, let χ ǫ : R → R be a smoothfunction such that χ ǫ is strictly convex for | t | < ǫ and χ ǫ = | t | for | t | ≥
0. Then wedefine a smooth maximum by g max ǫ ( x, y ) := x + y + χ ǫ ( x − y )2 . Note that g max ǫ ( x, y ) = max( x, y ) if | x − y | ≥ ǫ . Moreover, the smooth maximum of two C k -smooth (strictly) convex functions is C k -smooth (strictly) convex. Let dist( x, ∂ Ω) :=inf {k x − y k : y ∈ ∂ Ω } . Let δ : R n → R be the distance function of Ω defined by δ ( x ) := ( − dist( x, ∂ Ω) if x ∈ Ωdist( x, ∂
Ω) if x ∈ Ω ∁ . Define Ω ǫ := { x ∈ R n : δ ( x ) < ǫ } . Theorem 7.1.
Let Ω ⊂⊂ R n be a convex domain with C k ( k ≥ -smooth boundary.Then there exists a C k -smooth function ρ : R n → R satisfying ρ is a defining function of Ω . ρ is strictly convex on R n \ ∂ Ω . The author would like to acknowledge that he learned the formulation of this theoremas well as the proof from Professor N. Shcherbina.
Proof.
We may assume that 0 ∈ Ω. Consider the Minkowski function of Ω. Define afunction f : R n → R defined by f ( x ) = ( λ, where λ x ∈ ∂ Ω if x = 00 if x = 0 . Then f is well-defined, C k -smooth on R n \ { } , and ∇ f = 0 on ∂ Ω. Convexity of Ωimplies that f is a convex function. Consequently, σ := f − ǫ > ∈ Ω − ǫ , we claim that there exist δ , δ > δ k x k − δ < ∂ Ω, and δ k x k − δ > σ on Ω − ǫ . Let S ǫ := max { σ ( x ) : x ∈ Ω − ǫ } < m ǫ :=min {k x k : x ∈ Ω − ǫ } > M ǫ := max {k x k : x ∈ ∂ Ω } >
0. Then one may choosesufficiently small δ , δ > δ M ǫ < δ < δ m ǫ + ( − S ǫ ) . Therefore δ k x k − δ ≥ δ m ǫ − δ > S ǫ > σ ( x )on Ω − ǫ , and δ k x k − δ ≤ δ M ǫ − δ < ∂ Ω. The claim is proved.Let V ǫ := { x ∈ R n : σ − ( δ k x k − δ ) < } and m := min { dist( ∂V ǫ , ∂ Ω − ǫ ) , dist( ∂V ǫ , ∂ Ω) } .Define ρ ǫ := g max m ( σ, δ k x k − δ ) . Then ρ ǫ is a C k -smooth function on R n and ρ ǫ = δ k x k − δ on Ω − ǫ which is strictlyconvex. Assume that 0 ∈ Ω (the number 1 is not important here and one may choosea number smaller than 1). Let U ǫ := { x ∈ R n : | δ ( x ) | < ǫ } and for j ∈ N c j := sup U j sup ≤| α |≤ k (cid:12)(cid:12)(cid:12)(cid:12) ∂ α ∂x α ρ j ( x ) (cid:12)(cid:12)(cid:12)(cid:12) , η j := 12 j c j Here, we used multi-index α = ( α , · · · , α n ) ∈ N n and | α | = α + · · · + α n , ∂ α ∂x α = ∂ α ∂x α ··· ∂x αnn . Define σ ( x ) := ∞ X j =1 η j ρ j ( x ) . Then for all j ∈ N , 0 ≤ | α | ≤ k , since (cid:12)(cid:12)(cid:12) ∂ α ∂x α ρ j ( x ) (cid:12)(cid:12)(cid:12) ≤ j on U j and P ∞ j =1 12 j converges, ∞ X j =1 η j ∂ α ∂x α ρ j ( x )is uniformly convergent and hence continuous on R n . Note that ∇ σ = P ∞ j =1 η j ∇ ρ j = 0on ∂ Ω. Therefore σ is a well-defined C k -smooth function on R n which is strictly convex N THE STEINNESS INDEX 21 on Ω.Now, we consider the outside of Ω. The argument is similar to that above. There exist e δ , e δ > σ ( x ) + e δ k x k − e δ < ∂ Ω, and σ ( x ) + e δ k x k − e δ > ∂ Ω ǫ .Let e V ǫ := { x ∈ R n : σ + e δ k x k − e δ < } and e m := min { dist( ∂ e V ǫ , ∂ Ω ǫ ) , dist( ∂ e V ǫ , ∂ Ω) } .Then e ρ ǫ := g max e m (0 , σ ( x ) + e δ k x k − e δ )is C k -smooth function on R n which is strictly convex on Ω ∁ ǫ . Finally, for e η j > e η j ց ρ ( x ) := σ ( x ) + ∞ X j =1 e η j e ρ j ( x )is the desired defining function of Ω. (cid:3) Corollary 7.2.
For a convex domain Ω ⊂⊂ C n with C k ( k ≥ -smooth boundary, DF (Ω) = 1 and S (Ω) = 1 .Proof. By Theorem 7.1, there exists a defining function ρ of Ω which is strictly convex on C n \ ∂ Ω. Hence ρ is strictly plurisubharmonic on C n \ ∂ Ω. This implies that DF (Ω) = 1and S (Ω) = 1. (cid:3) Remark . If the boundary regularity of a convex domain is C ∞ -smooth, then thereis another way to prove Corollary 7.2. σ in the proof of Theorem 7.1 is C ∞ -smoothplurisubharmonic defining function of Ω. Hence the theorem by Fornæss and Herbig([7]) implies DF (Ω) = 1 and S (Ω) = 1. Acknowledgements : The author would like to express his deep gratitude to Pro-fessor Kang-Tae Kim for valuable guidance and encouragements, and to Professor N.Shcherbina for fruitful conversations.
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