Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 327–331

ON THE MOVEMENT OF THE POINCARÉ METRIC

WITH THE PSEUDOCONVEX DEFORMATION OF OPEN RIEMANN SURFACES

Takashi Kizuka

Kyushu University 36 (Faculty Eng.), Graduate School of Mathematics Fukuoka 812, Japan; kizuka@math.kyushu-u.ac.jp

Abstract. The movement of the Poincaré metrics of open Riemann surfaces belonging to an analytic family defined on a 2-dimensional complex manifold Ω is logarithmically plurisubharmonic in Ω if Ω is Stein. As a corollary, we get a theorem due to Nishino.

0. Introduction

It has been known since Riemann that differentiable surfaces having a common constant Gauss curvature k are locally isometric to each other. Hence, thanks to Gauss–Bonnet’s theorem, we know that a Riemann surface ℜ of non-exceptional type has the unique complete hermitian metric ds_{ℜ}^{2} with constant Gauss curvature k = −4, which we call the Poincaré metric of ℜ. Let ℜ̃ denote the universal covering surface of ℜ with the canonical projection π: ℜ̃ ⟶ ℜ. The induced metric π^{∗}ds_{ℜ}^{2} is the erstwhile Poincaré metric of ℜ̃, which is biholomorphically equivalent to the unit disc D.

Let Ω be a two-dimensional Stein manifold and let f be a holomorphic function defined on Ω such that df ≠ 0 at each point of Ω. We treat the foliation defined by prime surfaces (irreducible components of level surfaces) of f in this paper. Let S_{c} be a prime surface of f with value c and suppose that S_{c} is not of exceptional type. We denote the Poincaré metric of S_{c} by ds_{c}^{2} . In the case where S_{c} is of exceptional type, set ds_{c}^{2} ≡ 0 on S_{c}. We also call ds_{c}^{2} the Poincaré metric of S_{c} in the latter case. We prove that the movement of ds_{c}^{2} is logarithmically plurisubharmonic in Ω in the following sense: Each point of Ω has a neighborhood U and a holomorphic function g in U such that z = g(p), w = f(p) (p ∈ U) defines a biholomorphic mapping of U onto a domain of C^{2} . Suppose that the Poincaré metrics ds_{w}^{2} of prime surfaces S_{w} satisfying S_{w} ∩ U ≠ ∅ have the expression ds_{w} = A(z, w)|d(z | S_{w})| on S_{w} ∩ U with respect to the local holomorphic coordinate system (z, w). Then log A(z, w) must be a plurisubharmonic function in U. This assertion is independent of the choice of the function g.

This fact was first noted by H. Yamaguchi [4, Corollary 3] in 1981 for the special case that each level surface of f is biholomorphically equivalent to the unit disc and that the boundaries of Ω and S_{c} are smooth, where he has used Hadamard’s variational method. We prove this result generally and directly using a purely function-theoretic idea.

1991 Mathematics Subject Classification: Primary 32A10; Secondary 32G05.

1. Robin constant and Poincaré metric of the unit disc

Let ds^{2} denote the Poincaré metric of the unit disc D. Let z be a local holomorphic coordinate system around a point p of D such that z(p) = 0. Assume that ds^{2} has the expression

ds = A(z)|dz|

with respect to the local coordinate system z. Since k = −(Δ_{z} log A)/A^{2} = −4, log A is a subharmonic function on the variable z.

Let ζ be a standard holomorphic coordinate system of D such that ζ(p) = 0. Then A(z) = |dζ/dz|/(1 − |ζ|^{2}), and so A(0) = |dζ/dz|_{z=0} . Let g_{p} denote the Green function of D with pole at p. The Robin constant λ_{z} for (D, p) with respect to the local coordinate system z is the real number

lim_{z→0} g_{p}(z) + log |z|.

Since g_{p} = − log |ζ|, we get λ_{z} = log(|dz/dζ|_{ζ=0}). Therefore λ_{z} = − log A(0). The following result can now be easily proved.

Lemma 1.1. Let D_{j} (j = 1, 2, …) be a sequence of simply connected subdomains of the unit disc D such that D_{j} ⊂ D_{j+1} and D = ∪D_{j}. Then the sequence of the Poincaré metrics ds_{j}^{2} of D_{j} converges monotonously to the Poincaré metric ds^{2} of D.

2. The movement of ds_{c,α}^{2}

Let Ω be a two-dimensional Stein manifold. Suppose that there exists a holomorphic function f on Ω such that df ≠ 0 at each point of Ω. Fix a smooth strictly plurisubharmonic function ϱ in Ω such that Ω^{α} = {p ∈ Ω | ϱ(p) < α} is relatively compact in Ω for each real number α.

For a point p_{0} of Ω, fix a holomorphic function g in a relatively compact neighborhood U of p_{0} such that z = g(p), w = f(p) (p ∈ U) defines a biholomorphic mapping G of U onto a bidisc B = {(z, w) ∈ C | |z| < 1, |w−f(p_{0})| < ε} for some positive constant ε. Fix a real number α such that U ⊂⊂ Ω^{α}. Set O = G^{−1}({(z, w) ∈ B | z = 0}). Let c be a complex number satisfying |c − f(p_{0})| < ε. Let S_{c}^{α} denote the prime surface of f | Ω^{α} with value c which passes O, where f|Ω^{α} is the restriction of f to Ω^{α}, and ds_{c,α}^{2} the Poincaré metric of S_{c}^{α}. In this section, we prove that the movement of ds_{c,α}^{2} is logarithmically plurisubharmonic in U.

Set O_{c} = O ∩ S_{c}^{α}. Because of the subharmonicity of the restriction ϱ | S_{c} of ϱ to S_{c}, we get the following lemma due to T. Nishino [2].

Lemma 2.2. Let S_{c} denote the prime surface of f with value c which contains S_{c}^{α}. Let γ be a closed continuous curve on S_{c}^{α} beginning and ending at O_{c}. If γ is not null-homotopic on S_{c}^{α} with base point O_{c}, then γ is not null-homotopic on S_{c} with base point O_{c}.

Set a = f(p_{0}). Let S̈ be a domain in the prime surface S_{a} such that S_{a}^{α} ⊂⊂ S̈ ⊂⊂ S_{a}. We also get the following

Lemma 2.3. There exists a tubular neighborhood V of S̈ in Ω and a holomorphic mapping φ of V onto S̈ such that the mapping Φ: p ↦ (φ(p), f(p)) (p ∈ V ) maps V onto the direct product S̈×Γ biholomorphically where Γ = {c ∈ C | |c−a| < δ} for some positive number δ and such that S_{c}^{α} ⊂⊂ (S_{c} ∩V) ⊂⊂ S_{c} for each c ∈ Γ.

Proof. We prove this lemma using Nishino’s trick. Each point of Ω has a holomorphically convex neighborhood W with a holomorphic vector field X_{W} such that (X_{W})_{p}f = 1 for each point p in W. Since Ω is Stein, we can construct a global holomorphic vector field X on Ω which satisfies X_{p}f = 1 for each point p in Ω. The system of local solutions of the partial differential equation X_{p}g = 0 defines a transversal holomorphic foliation on Ω with the holomorphic foliation defined by the prime surfaces of f. It proves the lemma.

Let Ṽ denote the universal covering of the tubular neighborhood V of S̈ in Lemma 2.3 whose canonical projection we denote by ϖ: Ṽ → V . The analytic surface ϖ^{−1}(S_{c} ∩ V) is the universal covering surface of S_{c} ∩ V and the manifold Ṽ is biholomorphically equivalent to the direct product D × Γ. So we identify Ṽ with D × Γ hereafter. Fix a connected component U^{∗} of ϖ^{−1}(U ∩ V). Set 𝒟^{α} = ⋃_{c} S_{c}^{α}. Then 𝒟^{α} is a subdomain of V ∩Ω^{α} (cf. Nishino [1]). Let Š_{c}^{α} denote a connected component of ϖ^{−1}(S_{c}^{α}) which passes U^{∗}. Because of Lemma 2.2, each Š_{c}^{α} is a simply connected subdomain of D × {c}. Hence Š_{c}^{α} is the universal covering surface of S_{c}^{α} with the projection ϖ | Š_{c}^{α}. Set 𝒟̌ = ⋃_{c} Š_{c}^{α}, which is a subdomain of Ṽ. The manifold 𝒟̌ is an unramified covering of 𝒟^{α} and the section of 𝒟̌ by the complex line w = c is Š_{c}^{α}.

Let ξ be a standard holomorphic coordinate system of D. In the following, we treat the manifold Ṽ = D × Γ as a domain of the direct product P × Γ where P is the Riemann sphere equipped with the inhomogeneous coordinate system ξ. The subdomain 𝒟̌ of D × Γ is pseudoconvex in P × Γ since the frontier points of 𝒟̌ in D×Γ are strongly pseudoconvex. Let dš_{c}^{2} denote the Poincaré metric of Š_{c}^{α} which has the expression dš_{w} = Ǎ(ξ, w)|d(ξ | Š_{w}^{α})| with respect to the coordinate system (ξ, w) of 𝒟̌. It suffices for us to prove that log Ǎ(ξ, w) is plurisubharmonic in U^{∗} .

As is seen in the beginning of Section 1, log Ǎ(ξ, c) is a subharmonic function in U^{∗} ∩ Š_{c}^{α} for each constant c ∈ Γ. So, for a subdomain Γ′ of Γ, we prove that log Ǎ(ψ(w), w) is a subharmonic function on the variable w for an arbitrary holomorphic function ψ in Γ′ satisfying (ψ(w), w) ∈ U^{∗} for each w ∈ Γ′. Let λ_{ξ′}^{w} denote the Robin constant for (Š_{w}^{α}, (ψ(w), w)) with respect to the local coordinate system ξ′|Š_{w}^{α} where ξ′ is the meromorphic function ξ − ψ(w) defined on P × Γ′. Since Ǎ(ξ, w)|d(ξ|Š_{w}^{α})| = Ǎ(ξ′ + ψ(w), w)|d(ξ′|Š_{w}^{α})|, it follows from Section 1 that λ_{ξ′}^{w} = − log Ǎ(ψ(w), w). Set σ = {(ξ, w) ∈ P × Γ′ | ξ = ψ(w)}. Consider the mapping Ψ of (P × Γ′) − σ onto Γ′ × C defined by x = w(p), y = 1/ξ′(p) (p ∈ (P × Γ′) − σ). The complement K of the image Ψ(𝒟̌ − σ) in Γ′ × C is a pseudoconcave subset of Γ′ × C. Let K_{t} denote the section K ∩ L_{t} of K by the complex line L_{t} = {(x, y) ∈ Γ′ × C | x = t}. As H. Yamaguchi proved in 1971 by a function-theoretic deduction, the transfinite diameter d_{∞,t} of K_{t} is a logarithmically subharmonic function on the variable t (cf. Yamaguchi [5]). Thanks to G. Szegö [3], we know that λ_{ξ′}^{t} = − log d_{∞,t}. Hence we have proved that log Ǎ(ψ(w), w) is a subharmonic function on the variable w.

3. Conclusions

Since ds_{c,β} ≤ ds_{c,α} for real numbers α and β satisfying α < β, it is sufficient for the proof of the assertion in Introduction to prove that ds_{c,α} → ds_{c} (α → ∞). Let S̃_{c} denote the universal covering surface of S_{c} with the canonical projection π: S̃_{c} → S_{c}. Fix a point p̃ of π^{−1}(p_{0}). Let D_{c}^{α} denote the connected component of π^{−1}(S_{c}^{α}) which contains p̃. Because of Lemma 2.2, D_{c}^{α} is a simply connected domain of S̃_{c} and D_{c}^{α} ⊂ D_{c}^{β} for real numbers α and β satisfying α < β. Suppose that S_{c} is not of exceptional type. Since S̃_{c} = ⋃_{α} D_{c}^{α}, we get by Lemma 1.1 that ds_{c,α} → ds_{c} (α → ∞). When S_{c} is of exceptional type, we can prove easily that ds_{c,α} → 0 (α → ∞). Using a tubular neighborhood of S_{c}^{α}, we can prove by this fact that A(z, w) in Introduction is upper semi-continuous. Hence A(z, w) must be logarithmically plurisubharmonic by the result of the previous section.

Therefore we get the following

Theorem. Let f be a holomorphic function on a two-dimensional Stein manifold Ω such that df ≠ 0 at each point of Ω. Then the movement of the Poincaré metrics of prime surfaces of f is logarithmically plurisubharmonic in Ω.

Corollary (T. Nishino [2]). Let f be a holomorphic function on a twodimensional Stein manifold. Set e = {c ∈ C | at least one prime surface of f with value c is of exceptional type}. If the logarithmic capacity of e is not zero, then every prime surface of f is smooth and of exceptional type.

In the case where df ≠ 0 at each point, the proof of the above corollary is straightforward. For the general case, we must prove the fundamental lemma of T. Nishino [2] in a modified form to fit our situation. But the above theorem makes the proof of the modified fundamental lemma fairly easy.

On the movement of the Poincaré metric 331

References

[1] Nishino, T.: Nouvelles recherches sur les fonctions entières de plusieurs variables complexes (I). - J. Math. Kyoto Univ. 8, 1968, 49–200.

[2] Nishino, T.: Nouvelles recherches sur les fonctions entières de plusieurs variables complexes (III). - J. Math. Kyoto Univ. 10, 1970, 245–271.

[3] Szegö, G.: Bemerkungen zu einer Arbeit von Herrn M. Fékete. - Math. Zeit. 21, 1924, 203–208.

[4] Yamaguchi, H.: Calcul des variations analytiques. - Japan. J. Math. 7, 1981, 319–377.

[5] Yamaguchi, H.: Variation of domains over C^{n}. - Lecture Note in Brown Univ., 1988.

Received 14 February 1994