On a High Dimensional Riemann ’ s Mapping Theorem and Its Applications

We prove that the domain D in Γ×Cz where Γ is a polydisk centered at (0) and the fiber of D over every point of Γ is a simply connected domain in Cz which contains a small disk {|z| ε}, where ε is independent of every point of Γ, is biholomorophic to some complete Hartogs domain. And we give applications of the uniformization of some fiber spaces.


Introduction
As for the classical uniformization problems such as the Hilbert's 22nd problem of a high dimensional case, there are some big theories of Griffith (1971) and Bers (1976). But the problem is not solved completely in the case of the manifold yet. For its special case such as an uniformization of a fiber space, Nishino (1969) proved that a Stein holomorphic family of C-type covering Riemann surfaces over a complex plane over a disk is C 1bundle. Yamaguchi (1976) gave a simpler proof than that of Nishino and proved more simply with Maitani (2004) by another principle. In (Yamaguchi, 1976), he introduced a Robin-Yamguchi function λ(t) which is the Robin constant of the Green function of a fiber D t when the base complex parameter of t is fixed which has a pole at a point of D t which is independent to the parameter t , proved that log λ(t) is superharmonic under some conditions and solved an uniformization problem of a Stein holomorphic family of covering Riemann surfaces over P 1 by using this fact in the case that the genus of them is zero and they are Riemann surfaces of parabolic type. His theorem includes the Nishino's result as a special case.
The fiber spaces studied by Nishino and Yamaguchi was the one over a disk. Fujita (1979Fujita ( , 1980Fujita ( , 1987 extended their studies to fiber spaces over a polydisk and Yamaguchi and others also extended the Robin-Yamaguchi function of several complex variables and got many remarkable results especially in (Kim, Levenberg, & Yamaguchi, 2011).
But the original problem of an uniformization of a Stein holomorphic family whose fibers are of hyperbolic type Riemann surfaces has been unsolved after that time except for a special case as in (Browder & Yamaguchi, 1994).
In this note, we prove for a manifold D such as a fiber space D = (D, π, Γ) which is considered as a Riemann domain over Γ × P 1 where Γ is a polydisk centered at (0), which has some schlicht branch and whose fibers over Γ are simply connected ramified covering Riemann surfaces over P 1 is mapped biholomorphically to a complete Hartogs domain (Theorem 5.1). This problem is reduced to a high dimensional Riemann's mapping theorem (Theorem 4.4).
We give some applications to the uniformization problem of fiber spaces which have topologicaly the same type fibers (Theorem 5.2 and 5.3).
Finally we have a local uniformization of a Stein holomorphic family whose fibers are topologically same type open Riemann surfaces (Definition 6.1) by reducing to a Riemann domain which can be applied to a high dimensional Riemann mapping theorem (Proposition 6.5 and Theorem 6.6).

Proposition 2.5 Let D be the same of Proposition 2.4. Let f be a holomorphic function ofD and lim
Then the Robin constant of the Green function g of D which has the pole at z = 0 is log R |a| and f is a conformal map of D to {|w| < R}. Proof. Since g = log | R f | = − log |z| + log | z f |R is a Green function of D which has the pole at z = 0. It is easy to see that the Robin constant of g is log R |a| and f is a conformal map of D to {|w| < R} by argument principle. From Corollary 2.2 the following corollary is easy to see.
Corollary 2.6 If a = 1 in the above proposition, f = ϕ where ϕ is the same of Proposition 2.4.
From Proposition 2.4, 2.5 and Corollary 2.6 the following proposition is easy to see.
Proposition 2.7 Let D and ϕ be the same of Proposition 1.4. Let R 0 is a positive constant such as R 0 < R. If we set D 0 = ϕ −1 (|w| < R 0 ), then Robin constant of the Green function of D 0 which has the pole at z = 0 is log R 0 .

Preliminary(2)
Definition 3.1 We denote a domain H := {(t, w); |w| < R t ∞ for t ∈ Δ(ρ) where Δ(ρ) is a disk centered at 0 with radius ρ(0 < ρ ∞)}. We call H a complete Hartogs domain. We denote by H(R) if R t ≡ R. (In the case (t) varies in a polydisk Γ centered at (0) we call it samely.) Following proposition is well known essentially due to Hartogs.

Proposition 3.2 The complete Hartogs domain H is a Stein one, if and only if − log R t is subharmonic or identically
Proof. When there is such a map Φ, |w| = |z| R t and ∂ 2 ∂t∂t log |w| = ∂ 2 ∂t∂t log |z|− ∂ 2 ∂t∂t log R t = 0 when z 0, ∂ 2 ∂t∂t log R t = 0. When log R t is harmonic, we set log R t = u(t) where t = x + iy, x, y ∈ R and t ∈ Δ(ρ). And we set the 1-form ω = − ∂u ∂y dx + ∂u ∂x dy. Then ω satisfies the integrability condition because u(t) is a harmonic function on Δ(ρ). Then u(t) and v(t) satisfy Cauchy-Riemann differential equations. We set When the case (t) varies in a polydisk Γ, refer to (Nishino, 2001, p. 14).
we have a conclusion from Proposition 3.3. Remark 3.5 Let H be the complete Hartogs domain and R t is continuous function of t. Then H is homeomorphic to H(R) with 0 < R ∞.
Definition 3.6 Let D be a domain in Δ(ρ) × C z and for every fixed t 0 in Δ(ρ), D t 0 := D ∩ {t = t 0 } is a bounded simply connected domain surrounded by a C 2 smooth Jordan curve. And when D t {z = 0} for every t ∈ Δ(ρ) and We set We remark that it does not depend on the choice of defining function ψ(t, z) of t∈Δ(ρ) ∂D t = {ψ(t, z) = 0} with the property of the above definition.
Proposition 3.7 (cf. Shiffer (1946, pp. 417, 418) and also Theorem 3.1 in Maitani and Yamaguchi (2004)) Let D ∈ (A). Then the Green function of D t which has the pole at z = 0 (especially the Robin constant log R t of D t ) varies in C 2 class with respect to the variable t.
Lemma 3.8 (Theorem 3.1 in Maitani and Yamaguchi (2004)) Let D ∈ (A) and Φ be a map of D to H such that Φ: is a conformal map of D t to {|w| < R t } such that ϕ(t, 0) = 0 and ∂ϕ ∂z (t, 0) = 1. Then, for the Robin-Yamaguchi function log R t of D and for the Green function log R t |ϕ| , − log R t |ϕ| is extended to a defining function of t∈Δ(ρ) ∂D t such as D ∈ (A) and the following equality is concluded.
ds is the arc length element of ∂D t and k 2 is above definition which means the Levi-curvature of t∈Δ(ρ) ∂D t .
Corollary 3.9 Let the situation be above and D be a Stein domain. Then

is a superharmonic function on D and H is a complete Hartogs Stein domain.
Proof. The first statement is easy to see from Lemma 3.8 and k 2 0 because D is a Stein domain. The second one is a well known fact when the first statement is true. The third one is followed by Proposition 3.2.
Following proposition is owed essentially to H. Yamaguchi.

Proposition 3.10 Let D be a Stein domain, D ∈ (A) and the Robin-Yamaguchi function log R t be harmonic on Δ(ρ).
Then D is Levi-flat (k 2 ≡ 0) and Φ is biholomorphic map from D to H where Φ is the same in Lemma 2.8.
Proof. By Lemma 3.8 and the assumption such as ∂ 2 ∂t∂t log R t = 0 and k 2 0, k 2 ≡ 0 and | ∂ 2 ∂t∂z log R t |ϕ| | = 0 when z 0. As log R t |ϕ| is a real valued function, ∂ 2 ∂t∂z log R t |ϕ| = 0 when z 0. Then ∂ 2 ∂t∂z log |ϕ| = 0 when z 0. Since z)) is holomorphic function of variables t and z separately and it is holomorphic by Hartogs theorem. Then Φ is biholomorphic.

Δ(ρ) if and only if there is a biholomorphic map
Proof. When D is a Stein domain and log R t is harmonic on Δ(ρ), D is Levi-flat and Φ: D → H is a biholomorphic map by Proposition 3.10 where Φ is the same in Lemma 3.8 and there is a map Φ 0 as above by Proposition 3.3.

Conclusion
Lemma 4.1 Let D be a domain of in Δ(ρ) × C z such that when an arbitrary number t 0 in Δ(ρ), D t 0 = D ∩ {t = t 0 } is a simply connected domain and D t 0 ⊃ {|z| ε} where ε is a positive constant which is independent of t 0 . We asumme that Φ is the map of D to H such as t = t, w = ϕ(t, z), which is a conformal map of D t to {|w| < R t ∞} such as ϕ(t, 0) = 0 and ∂ ∂z ϕ(t, 0) = 1 for fixed t. Then there is a constant δ such that a domain H(δ) with δ < R t for every t ∈ Δ(ρ) and Φ −1 (H(δ)) = D 0 is a domain of the class (A) and ϕ(t, z) is holomorphic on D 0 .
Proof. There is a small constant δ above by the modified Koebe one-quarter theorem. From Proposition 2.7 the Green function g 0 of D 0 which has the pole at z = 0 is log δ |ϕ| . From Proposition 3.7 −g 0 = log |ϕ| δ is a defining function of D 0 if V = D = Φ −1 (H(δ )) where δ and δ are nealy equal and δ < δ and D 0 ∈ (A) of Definition 3.6.
Set ψ = log |ϕ| δ and ϕ(t, z) = δ·z·exp(−k(t, z)) anew. Since ψ is a real valued function, ∂ψ ∂t = 1 2 ∂ϕ ∂t Then v(t, (x, y)) is a harmonic function of a variable t since ∂ 2 ∂t∂t (k −k) = ∂ 2 ∂t∂t (2iv(t, (x, y))) = 0 by above equation . And u(t, (x, y)) is a harmonic one also because As u(t, (x, y)), v(t, (x, y)) are harmonic functions of a variable t, ∂ 2 k ∂t∂t = ∂ ∂t ( ∂k ∂t ) = 0 and ∂k ∂t is a holomorphic function of a variable t. Therefore k(t, z) is a holomorphic one of t. By Hartogs theorem k is a holomorphic function of (t, z) because it is holomorphic with respect to t and z separately. Then ϕ(t, z) is a holomorphic function on D 0 .
Theorem 4.2 In the same situation of above lemma, the map Φ is biholomorphic.
Proof. Since ϕ(t, z) is holomorphic on D 0 and holomorphic of a variable z when t is fixed in Δ(ρ), ϕ(t, z) is a holomorphic function on D by virtue of Hartogs theorem.
From Theorem 4.2, Proposition 3.2 and 3.3, the following corollary is easy to see.

Corollary 4.3 Let D be the same of Lemma 4.1. Then D is a Stein domain if and only if − log R t is subharmonic or identically −∞ on Δ(ρ). And D is biholomorphic to H(1) if and only if − log R t is harmonic on Δ(ρ).
Theorem 4.4 Let D be a domain in Γ × C z where Γ := {(t 1 , . . . , t n ); |t i | < ρ i , 0 < ρ i ∞, i = 1, . . . , n} and for every point (t 0 ) ∈ Γ, D (t 0 ) := D ∩ {(t) = (t 0 )} is a simply connected domain and D (t 0 ) ⊃ {|z| ε} for every (t 0 ) ∈ Γ where ε is a positive number which is independent of (t 0 ). Let log R (t) be the Robin-Yamaguchi function of D (t) .
Then the map Φ of D to a complete Hartogs domain H such as (t) = (t), w = ϕ((t), z), which is a conformal map of D (t) to {|w| < R (t) ∞} with fixed (t) such as ϕ((t), 0) = 0 and ∂ ∂z ϕ((t), 0) = 1, is biholomorphic. Proof. If we fix (t 1 , . . . , t n ) except t i (1 i n) and fix z ∈ {|z| ε}, ϕ is holomorphic with respect to t i by Theorem 4.2. If we fix (t), ϕ is holomorphic with respect to z. Then ϕ is holomorphic on Γ × C z by Hartogs theorem. It is easy to see that Φ is biholomorphic.
From Theorem 4.4, Proposition 3.2 and 3.3, the following corollary is easy to see.

Applications
Following theorem is easy to see by the same method of Theorem 4.4 and the uniformization theorem by Koebe.
Theorem 5.1 Let D = (D, π, Γ) be a connected manifold such as a ramified Riemann domain over Γ × P 1 with www.ccsenet.org/jmr Journal of Mathematics Research Vol. 6, No. 3;2014 the projection π where Γ is the same of Theorem 4.4 and it has a schlicht branch D 0 over Γ × {|z| ε} for some ε > 0. And we assume that for every (t) ∈ Γ, the fiber D (t) is an irreducible ramified covering Riemann surface such as simply connected. Then D is biholomorphic to some complete Hartogs domain H in Γ × {|w| ∞} with fiber preserving.
Theorem 5.2 Let D is the same of Theorem 5.1. We assume that D is a Stein manifold and for every (t) ∈ Γ, the fiber D (t) is an irreducible ramified covering Riemann surface with genus g( ∞) and the number of boundary components is n(1 n ∞) where g and n are independent of (t). We call such D of (g, n) type.
Then we have the following results (I) and (II): (I) If (g, n) = (0, 1), D (t) is simply connected and there are following cases: (a) When for every (t) ∈ E which is a capacity positive set with E ⊂ Γ, D (t) is holomorphically isomorphic to C, X is biholomorphic to H(∞).
(b) When for every (t) ∈ Γ except for a capacity zero set, D (t) is holomorphically isomorphic to the unit disk, D is biholomorphic to a complete Hartogs Stein domain H (The definition of the set of capacity zero or positive, see Fujita, 1987, p. 685).
(b ) When for every (t) ∈ Γ, D (t) is holomorphically isomorphic to the unit disk, D is biholomorphic to H with R t < ∞ and D = (π −1 (Γ ), π, Γ ) where Γ is the arbitrary polydisk centered at (0) such as Γ Γ is biholomorphic to some bounded complete Hartogs Stein domain H .
(II) If (g, n) (0, 1) and X is homeomorphic to Γ × R where R is a Riemann surface of (g, n) type, the universal covering space D of D with base point ((t), o) ∈ D 0 where o is the point in D 0 t whose projection is z = 0, D is homeomorphic to Γ × R and there are following cases: (c) When D (t) is the case (a) of (I), that is D (t) is holomorphically isomorphic to C * for a capacity positive set E ⊂ Γ, D is biholomorphic to H(∞).
(d) When D (t) is the case (b) of (I), D is biholomorphic to H as the same of (b) of (I).
(d ) When D (t) is the case (b ) of (I), D is biholomorphic to H with R t < ∞ and D is biholomorphic to some bounded complete Hartogs Stein domain H which is samely defined as above (b ) of (I).
By Theorem 4.4, ϕ((t), p) is holomorphic on D 0 . By Hartogs theorem, (t, ϕ) is biholomorphic from D to H. Since H is a complete Hartogs Stein domain of Γ×C, − log R (t) is plurisubharmonic or identically −∞ by Proposition 3.2. When − log R (t) = −∞ for a capacity positive set E ⊂ Γ, − log R (t) ≡ −∞ on Γ by the nature of plurisubharmonic function and the case (a) occurs. When − log R (t) −∞, − log R (t) > −∞ except for a capacity zero set of Γ and the case (b) occurs. When − log R (t) > −∞ on Γ, − log R (t) M on Γ where M is a real constant and the case (b ) occurs.
The case (c), (d) and (d ) of (II) is easy to see from the above discussion because D is a Stein manifold and biholomorphic to a complete Hartogs Stein domain.
Theorem 5.3 Let D = (D, π, Γ) be a manifold such as a ramified Riemann domain over Γ × P 1 and for every (t) ∈ Γ, the fiber D (t) is a compact Riemann surface with genus g( 1) which is independent of (t) where Γ is the same of Theorem 4.4. And it has a schlicht branch D 0 such as over Γ × {|z| ε, ε > 0}.
Then D which is constructed as the same way of Theorem 5.2 (II), is biholomorphic to Γ × C when g = 1 and D is biholomorphic to a complete Hartogs domain H when g 2 where the Hartogs radious R (t) < ∞ for every (t) ∈ Γ.
Proof. When g = 1, D is in the case (I) (a) of Theorem 5.2 and when g 2, D (t) is holomorphically isomorphic to the unit disk, above statement is easy to see from Theorem 5.1.

and
(3) (D, π, Γ) is homeomorphic to Γ × R preserving fibers where R is an open Riemann surface.
Then we call D = (D, π, Γ) a Stein holomorphic family of open Riemann surfaces with respect to R.
Proposition 6.2 Let D be the same above. Then the canonical bundle K D has a glovbal holomorphic section g which has zero only on S with any high order. If S = ∅, we take g such as g 0. Proof. We consider the linear differential equation ∂( f 1 ,··· , f n ,u) (1) where g is the same of Proposition 6.2, (v 1 α , · · · , v n+1 α ) are any coordinate variables of D and u is an unknown function. By virtue of the Cauchy-Kovalevskaya theorem, there is a neighborhood W of a point p in D − S where π(p) is an arbitrary point of Γ and a local holomorphic solution u W in W of (1) where u W | σ = 0 where σ is a local holomorphic section of D through p. By the same method of Lemma 2.2 in (Adachi, 2001) the equation (1) can be regarded as an analytic family of holomorphic 1-forms on fibers D (t) = { f 1 = t 1 , · · · , f n = t n } which can be considered as characteristic curves of (1) because we can take g which has zero only on S with sufficient high order if the integer m in the proof of above proposition take sufficient larage. The local solution u W has an analytic continuation along each fiber endlessly through W and u W is single valued holomorphic function on (π −1 (W), π, π(W)) and a global holomorphic solution u ∈ O(D) from Lemma 2.2 in (Adachi, 2001).
Set F = ( f 1 , . . . , f n , u). Then it is a holomorphic map from D to Γ × C. It is easy to see from the condition (2) of Definition 5.1 that F is scatterd inverse map. Then from Osgood theorem, F −1 defines a branched Riemann domain over Γ × C which is biholomorphic to D. When S = ∅, it is biholomorphic to an unbranched Riemann domain over Γ × C because we can take g such as g 0 from Proposition 6.2.
Proposition 6.4 If R is not simply connected, the universal covering Stein manifoldD = (D,π, Γ) which is homeomorphic to Γ ×R is biholomorphic to a branched Riemann domain over Γ × C. When S = ∅,D is biholomorphic to an unbranched Riemann domain over Γ × C.
Proof. We consider the eqation (1) in the proof of Proposition 6.3. We use the same notations in it. The local holomorphic solution u W in W of (1) where u W | σ = 0 has an analytic continuation along any path on each fiber. Since the universal covering spaceD is constructed by takeing the universal covering of every fiber, F is a holomorphic map fromD to Γ × C becauseD is a Stein manifold and F −1 defines a branched Riemann domain over Γ × C which is biholomorphic toD. When S = ∅, it is easy to see the conclusion by the same reason of the proof of above proposition and above discussion.
Proof. Since the holomorphic map F :D → Γ × C in the proof of above proposition such as F| σ = 0 and Jacobian of F is not 0 near Γ × {0}, it is easy to see the conclusion above when we take Γ small again if necessaly.
Theorem 6.6 Let D = (D, π, Γ) be a Stein holomorphic family of open Riemann surfaces with respect to an arbitrary Riemann surface R. For every (t) ∈ Γ there is a neighborhood v(t) Γ such thatD = ( π −1 (v), π , v) is biholomorphic to to some Hartogs Stein domain H in γ × C where γ is a polydisk centered at (0) which is biholomorphic to v(t).