UNIFIED APPROACH TO CONFORMALLY INVARIANT METRICS ON RIEMANN SURFACES

TOSHIYUKI SUGAWA

Abstract. In this article, we will survey the relationship between various conformally invariant (pseudo-)metrics on Riemann surfaces and classical extremal problems. As a simple application, we will give universal estimates for (pseudo-)metrics which satisfy a contraction property with respect to a certain class of holomorphic maps.

1. Introduction

In the unit disk 𝔻 = {z ∈ ℂ; |z| < 1}, the Poincaré metric |dz|/(1 − |z|^{2}) plays quite an important role in modern function theory. Various generalizations of this metric to Riemann surfaces have been given, e.g., hyperbolic (Kobayashi) metric, Carathéodory metric, Hahn metric, Begman metric and so on. In this article, we will provide a unifying treatment of such invariant metrics and explain a general principle dominating those metrics by some universal metrics.

A family of conformal pseudo-metrics ρ_{R} defined for all Riemann surfaces R is said to be holomorphically contractive if f^{∗}ρ_{R′} ≤ ρ_{R} holds for any holomorphic map f : R → R′ and is said to be normalized if ρ_{𝔻}(0) = 1 holds for the unit disk and its canonical coordinate z. The following result is well known (cf. [5]).

Theorem 1.1. Any normalized, holomorphically contractive, conformal pseudo-metric ρ satisfies c_{R} ≤ ρ_{R} ≤ k_{R} for all Riemann surfaces R, where c_{R} and k_{R} denote the Carathéodory and the Kobayashi pseudo-metrics, respectively.

Many invariant metrics need not be holomorphically contractive but satisfy a weaker contraction property. A family of conformally invariant, conformal pseudo-metrics ρ_{R} is said to be monotone if ρ_{R_{0}} ≥ ρ_{R} holds for each Riemann surface R and for each subdomain R_{0} of R.

Theorem 1.2. Any normalized, conformally invariant, monotone, conformal metric ρ satisfies a_{R} ≤ ρ_{R} ≤ h_{R} for all Riemann surfaces R, where a_{R} and h_{R} denote the Ahlfors-Beurling and the Hahn pseudo-metrics, respectively.

Precise definitions and fundamental properties for the above metrics will be given in Sections 2 and 3. Section 2 will be devoted to the formal definition of (conformal) pseudometircs and explanations of principles generating invariant pseudo-metrics from various classical function spaces. We will state also a more general result about pseudo-metrics with some contraction property. As corollaries, we will show the above two theorems in Section 3, where we will provide characterizations of many classical (pseudo-)metrics in our context. A similar approach can be found in the book [10] by Sario and Oikawa.

1991 Mathematics Subject Classification. Primary 30F45; Secondary 30C40.

Key words and phrases. hyperbolic metric, Carathéodory metric, analytic capacity .

The author was partially supported by the Ministry of Education, Grant-in-Aid for Encouragement of Young Scientists, 11740088.

Some of the formulations here can be generalized to the case of several complex variables although many of them would only be Finsler metrics instead of Riemannian metrics, but we will not pursue it here.

2. Principles

First, we recall the definition of differential forms on Riemann surfaces. Let R be a Riemann surface with an atlas A consisting of local coordinates α : U_{α} → V_{α} from an open set U_{α} of R onto an open set V_{α} of ℂ such that ⋃_{α∈A} U_{α} = R and that β ∘ α^{−1} : α(U_{α} ∩ U_{β}) → β(U_{α} ∩ U_{β}) is holomorphic for each α and β in A.

Let m and n be half integers, i.e., 2m, 2n ∈ ℤ, with m + n being an integer. A family (ω_{α} : V_{α} → ℂ)_{α∈A} of functions is said to be an (m, n)-form on R and will be denoted conventionally by ω or ω_{α}(z)dz^{m}dz̄^{n} if for two local coordinates z = α(p) and w = β(p) in A these functions satisfy the transition relations

ω_{β}(w)(w′)^{m}(\overline{w′})^{n} = ω_{α}(z)

for z ∈ α(U_{α} ∩ U_{β}), where w = β ∘ α^{−1}(z) and w′ = dw/dz and the branch of w′^{1/2}\overline{w′}^{1/2} should be chosen to equal |w′| in the case that m and n are not integers. A differential form ω is said to be, for example, continuous or holomorphic if any coefficient ω_{α} is continuous or holomorphic. An (m, 0)-form is called an m-form or m-differential. If no confusion can occur, we often write ω = ω(z)dz^{m}dz̄^{n} for simplicity. When R is a plane domain, we will take the canonical coordinate z as a local coordinate unless a specification will be stated. A (0, 0)-form ω is nothing but a function on R, so we can write ω = ω(p) for p ∈ R without refering to any local coordinate. If all coefficients ω_{α} of an (m, m)-form are positive (or non-negative) for each local coordinate α, we will say that ω is positive (or non-negative). A positive (or non-negative) (\frac{1}{2}, \frac{1}{2})-form on R will be called a conformal metric (or conformal pseudo-metric) on R. In this article, a (pseudo-)metric will always be conformal. We note finally that a non-negative (1, 1)-form ω = ω(z)|dz|^{2} can be regarded as an area element on the surface under the identification dzdz̄ = |dz|^{2} = dx ∧ dy, where z = x + iy.

Let f : R → R′ be a holomorphic map. Then we can define the pullback of an (m, n)- form ω on R′ by f, which will be denoted by f^{∗}ω, by

(f^{∗}ω)_{α}(z) = ω_{β}(w)(w′)^{m}(\overline{w}′)^{n},

where α and β are local coordinates of R and R′, respectively, such that f(U_{α}) ⊂ U_{β} and w = β ∘ f ∘ α^{−1}(z).

Let R denote the category of all (connected) Riemann surfaces with holomorphic maps between them as morphisms. A family ρ = (ρ_{R}) of pseudo-metrics ρ_{R} on R for all Riemann surfaces R ∈ R will be simply called a pseudo-metric. A pseudo-metric ρ is said to be conformally invariant if f^{∗}ρ_{R′} = ρ_{R} holds for all conformal isomorphism f : R → R′. A conformally invariant pseudo-metric ρ is said to be normalized if ρ_{𝔻}(0) = 1 for the unit disk 𝔻. By conformal invariance, this means also that ρ_{𝔻} = |dz|/(1−|z|^{2}) for the canonical coordinate z of 𝔻.

Let C mean a class of holomorphic maps between Riemann surfaces, namely, C(R, R′) is a set of holomorphic maps from R to R′ for any pair of Riemann surfaces. A pseudo-metric ρ is said to be C-contractive if f^{∗}ρ_{R′} ≤ ρ_{R} holds for any f ∈ C(R, R′).

In the sequel, we will use mainly the next three classes:

O(R, R′) = {f : R → R′ holomorphic},
S(R, R′) = {f ∈ O(R, R′); injective or constant}, and
I(R, R′) = {f : R → R′ biholomorphic}.

Note that I(R, R′) is empty if R and R′ are not holomorphically equivalent.

By definition, ρ is I-contractive if and only if ρ is conformally invariant. Similarly, ρ is O-contractive if and only if ρ is holomorphically contractive, and ρ is S-contractive if and only if ρ is conformally invariant and monotone.

We now give an idea to systematically construct conformally invariant pseudo-metrics from the natural classes of functions on Riemann surfaces in the spirit of Ahlfors-Beurling [1].

Let m be a positive integer and C be a class of holomorphic maps between Riemann surfaces. A class F = (F(R))_{R∈R} of subsets of holomorphic m-forms on R will be called C-admissible if the following conditions are fulfilled:

(i) if f ∈ C(R, R′) then f^{∗}(F(R′)) ⊂ F(R), and (ii) the set F(R) is compact in the sense that any sequence in F(R) has a subsequence

converging to some element of F(R) uniformly on each compact subset of R. Given a C-admissible class F of holomorphic m-forms, we can define the pseudo-metric

ρ^{F} by

(ρ_{R,α}^{F}(z))^{m} = sup{|φ_{α}(z)|; φ ∈ F(R)},

where z = α(p) is a local coordinate of a Riemann surface R. In fact, for each point p ∈ R, by property (ii), we see that there exists an element φ ∈ F(R) such that ρ_{R}^{F} = |φ|^{1/m} at p ∈ R, in particular, ρ_{R}^{F}(p) < +∞. We then have the following result, a prototype of which can be found in [1, Theorem 1].

Proposition 2.1. If F is C-admissible, then the pseudo-metric ρ^{F} is continuous and C-contractive. Moreover, log ρ_{R}^{F} is subharmonic on R unless ρ_{R}^{F} = 0.

Remark. We say that log ρ_{R} is subharmonic if log ρ_{R,α} is subharmonic on V_{α} for each local coordinate α. Note that subharmonicity of log ρ_{R} is conformally invariant because log |dw/dz| is harmonic for a biholomorphic transition function w = w(z).

Proof. By definition, log ρ_{R,α}^{F} = \frac{1}{m} sup_{φ∈F(R)} log |φ_{α}| and each log |φ_{α}| is subharmonic in V_{α}. So the latter part of the proposition is now clear. In particular, ρ_{R}^{F} is lower semicontinuous. In order to show the upper semi-continuity of ρ_{R}^{F}, it is sufficient to see that ρ_{R}(p) ≥ α for a sequence p_{n} converging to p in R such that ρ_{R}(p_{n}) ≥ α for all n. Let φ_{n} ∈ F(R) be an extremal differential for which ρ_{R}(p_{n}) = |φ_{n}(p_{n})| holds. Then, by compactness of F(R), we may assume that the sequence φ_{n} converges to an element φ of F(R) uniformly on each compact subset of R. Therefore, we have ρ_{R}^{F}(p) ≥ |φ(p)| ≥ α. Finally, C-contractivity is straightforward to see.

We now explain a construction of pseudo-metrics which is dual with the above in some sense.

The complex tangent space T_{p}R of a Riemann surface R at a point p ∈ R is defined as the set of derivations from the space of holomorphic germs Hol_{p}(R) at p into ℂ, where a map v : Hol_{p}(R) → ℂ is called a derivation if v is linear and satisfies v(fg) = g(p)v(f) + f(p)v(g) for all f, g ∈ Hol_{p}(R). Let f : 𝔻 → R be a holomorphic map with f(0) = p. We can then assign to f a tangent vector v_{f} defined by

v_{f}(g) = \frac{d}{dz} (g ∘ f)(z)|_{z=0} (g ∈ Hol_{p}(R)).

Note that v_{f} is nothing but the image of the canonical tangent vector (d/dz)|_{z=0} of T_{0}𝔻 under the tangent map f_{∗} of f.

A class C of holomorphic maps between Riemann surfaces will be called ample if the following four conditions are satisfied:

(a) for each point p of an arbitrary Riemann surface R there exists a non-constant map f ∈ C(𝔻, R) such that f(0) = p,

(b) each f ∈ C(R, R′) satisfies f^{∗}(C(R′, 𝔻)) ⊂ C(R, 𝔻),

(c) for every Riemann surface R, C(R, 𝔻) is compact in the sense that any sequence in C(R, 𝔻) converging uniformly on compacta has its limit in C(R, 𝔻), and

(d) id_{𝔻} ∈ C(𝔻, 𝔻).

We set C_{p}(R) = {f ∈ C(𝔻, R); f(0) = p}. We now define a pseudo-metric σ^{C} by

σ_{R,α}^{C}(z)^{−1} = sup{|v_{f}(α)|; f ∈ C_{p}(R)},

where α is a local coordinate of R with z = α(p). Note that the ampleness guarantees σ_{R} < +∞. The pseudo-metric σ^{C} is actually normalized and C-contractive as is easily seen by property (d) the Schwarz lemma.

Remark. Formally, we may think that a tangent vector is a −1-form, so the above definition agrees with the former one for ρ^{F} in a formal sense.

We can also associate a pseudo-metric with a class C. Set

C^{♯}(R) = {df; f ∈ C(R, 𝔻)}

for an arbitrary Riemann surface R. Then C^{♯} is C-admissible by properties (b) and (c). So, we now set τ^{C} = ρ^{C^{♯}}. Then the pseudo-metric τ^{C} is normalized and C-contractive. Remember that τ_{R}^{C}(p) = sup_{f∈C(R,𝔻)} |df|(p) by definition.

The pseudo-metrics σ^{C} and τ^{C} are the minimum and the maximum among normalized C-contractive pseudo-metrics ρ, respectively.

Theorem 2.2. Let C be an ample class of holomorphic maps between Riemann surfaces. A normalized, C-contractive pseudo-metric ρ satisfies τ_{R}^{C} ≤ ρ_{R} ≤ σ_{R}^{C} for each Riemann surface R.

Proof. Fix a point p_{0} in R and a local coordinate α around p_{0}. Set z_{0} = α(p_{0}). Let f ∈ C^{♯}(R) = C(R, 𝔻) and put f_{α} = f ∘ α^{−1}. By assumption, we have

|f′_{α}(z_{0})| ≤ \frac{|f′_{α}(z_{0})|}{1 − |f_{α}(z_{0})|^{2}} = (f^{∗}ρ_{𝔻})_{α} ≤ ρ_{R,α}(z_{0}).

This implies τ_{R,α}^{C}(z_{0}) ≤ ρ_{R,α}(z_{0}).

We next take an f ∈ C_{p_{0}}(R) and set f_{α} = α ∘ f on f^{−1}(U_{α}). Then

ρ_{R,α}(z_{0})|f′_{α}(0)| = f^{∗}ρ_{R}(0) ≤ ρ_{𝔻}(0) = 1.

Noting f′_{α}(0) = v_{f}(α), we obtain ρ_{R,α}(z_{0}) ≤ σ_{R,α}^{C}(z_{0}).

3. Examples

In this section, we will demonstrate various classical invariant metrics can be re-defined in our framework. In this section, a metric, say ρ, is sometimes called metrics instead of pseudo-metrics when ρ_{R} is actually a metric unless ρ_{R} identically vanishes.

Example 1 (Kobayashi pseudo-metric k_{R}). The pseudo-metric k := σ^{O} is called the Kobayashi pseudo-metric (cf. [5]). By definition, we can easily see that this metric is holomorphically contractive. Furthermore, we have the next result.

Lemma 3.1. If R is hyperbolic, then k_{R} coincides with the hyperbolic metric, that is, π^{∗}k_{R} = |dz|/(1 − |z|^{2}) for a holomorphic universal covering π : 𝔻 → R of R. Otherwise, namely, R is conformally equivalent to either the Riemann sphere ℂ̂, the complex plane ℂ, the punctured complex plane ℂ^{∗} = ℂ − {0} or a complex torus (an elliptic curve), we have k_{R} = 0.

Proof. Suppose that R is hyperbolic and a point p ∈ R is given. Let α be a local coordinate with α(p) = z_{0}. Then we can take a holomorphic universal covering π from the unit disk with π(0) = p. It suffices to show that k_{R,α}(z_{0}) = 1/|v_{π}(α)|. Let f ∈ O_{p}(R). We can then take a lift f̃: 𝔻 → 𝔻 of f such that f̃(0) = 0 via the universal covering π. The Schwarz lemma now implies |v_{π}(α)/v_{f}(α)| = |f̃′(0)| ≤ 1, which means that k_{R,α}(z_{0})^{−1} = |v_{π}(α)|.

The latter part is obvious because we can take a holomorphic immersion f of ℂ into R with f(0) being equal to a given point in R.

Example 2 (Carathéodory pseudo-metric). The pseudo-metric c = τ^{O} is called the Carathéodory (-Reiffen) pseudo-metric. By definition, this pseudo-metric is normalized and holomorphically contractive. We directly see that c_{R} = 0 if and only if R ∈ O_{AB}, namely, R carries no non-constant bounded analytic functions.

The quantity c_{R}(p) is sometimes called the analytic capacity. An extremal function f : R → 𝔻 satisfying |df|(p) = c_{R}(p) is usually called the Ahlfors function at p and known to be unique up to unimodular constants (see [4]). We remark that the condition c_{R}(p) = 0 at some point p need not imply that c_{R}(p) = 0 at every point p in the case that R is non-planar. A counterexample was constructed by Virtanen [13] (see also [10, X. 2K]). The same can be said to the span metric s_{R} defined below.

Applying C = O to Theorem 2.2, we now have Theorem 1.1.

Example 3 (Hahn metric). The pseudo-metric h := σ^{S} is called the Hahn metric (see

[6] for details). This metric is normalized, conformally invariant and monotone. It follows that h_{R} = 0 if and only if R is conformally equivalent to either ℂ̂ or ℂ.

This metric is greatly useful because of conformal invariance and comparability with the quasi-hyperbolic metric in the case of plane domains.

Lemma 3.2 ([6]). For a proper subdomain D of ℂ, we have the estimate

\frac{1}{4δ_{D}(z)} ≤ h_{D}(z) ≤ \frac{1}{δ_{D}(z)},

where δ_{D}(z) = inf{|z − a|; a ∈ ∂D}.

Example 4 (Ahlfors-Beurling pseudo-metric). The pseudo-metric a := τ^{S} has been used in the paper [1] by Ahlfors and Beurling. This is normalized, conformally invariant and monotone. Note that a_{R} = 0 if and only if R is either of positive genus or planar and the boundary is of class N_{SB} = N_{SD}. For other characterizations of the property a_{R} = 0 for plane domains R, see [10, VII. 5C].

Applying C = S to Theorem 2.2, we have Theorem 1.2.

Example 5 (Bergman metric). Let Ω(R) be the space of square integrable holomorphic 1-forms ω on a Riemann surface R with norm

∥ω∥_{Ω(R)} = (\frac{i}{2} ∫∫ _{R} ω ∧ ω̄)^{1/2} = (∫∫ _{R} |ω(z)|^{2}dxdy)^{1/2} .

We define the class W by W(R) = \{ω ∈ Ω(R); ∥ω∥_{Ω(R)} ≤ \sqrt{π}\}. Then the pseudometric b := ρ^{W} is called the Bergman metric. This metric is also normalized, conformally invariant and monotone. Note that b_{R} = 0 if and only if R is planar and R ∈ O_{G} (see [10]). Virtanen [13] showed that if b_{R}(p) = 0 at some point p ∈ R then b_{R} identically vanishes (see also [10, II. 3C]).

The space Ω(R) is actually a Hilbert space with the inner product

(ω, δ)_{R} = \frac{i}{2} ∫∫ _{R} ω ∧ δ̄ = ∫∫ _{R} ω(z)\overline{δ(z)}dxdy.

For a local coordinate β of R, the linear functional ω ↦ ω_{β}(w) is bounded on Ω(R) for each w ∈ V_{β}, and hence there exists an element κ^{β,w} ∈ Ω(R) such that ω_{β}(w) = (ω, κ^{β,w})_{R} for all ω ∈ Ω(R) by the Riesz representation theorem. For another local coordinate z = α(p), we write K_{R,α,β}(z, w) = κ_{α}^{β,w}(z) for (z, w) ∈ V_{α} × V_{β}. For brevity, we sometimes write K_{R}(p, q) = K_{R}(z, w)dzdw̄ and call it the Bergman kernel of R. Note that \overline{K_{R}(p, q)} = K_{R}(q, p) and K_{R}(w, w) = (K_{R}(·, w), K_{R}(·, w))_{R} = ∥K_{R}(·, w)∥_{Ω(R)}. The Schwarz inequality implies that |ω(w)| = |(ω, K_{R}(·, w))_{R}| ≤ \sqrt{K_{R}(w, w)}∥ω∥_{R} with equality being valid if ω is a constant multiple of K_{R}(·, w). It then follows that b_{R}(w) = \sqrt{πK_{R}(w, w)}.

The reader should be careful of the fact that the Bergman (pseudo-)metric sometimes refers to the Hermitian form ∑_{jk} ∂_{z_{j}}∂z̄_{k} log K_{R}(z, z)dz_{j}dz̄_{k} in the theory of several complex variables.

The following proposition was proved by Suita [12]. (For finite Riemann surfaces, Hejhal proved it earlier.)

Proposition 3.3. The inequality c_{R} ≤ b_{R} holds for every Riemann surface R.

Example 6 (span pseudo-metric). Define the space Ω_{e}(R) as the closed subspace of Ω(R) consisting of exact differentials with the same norm as Ω(R). Set W_{e} = Ω_{e}(R) ∩ W. Then the pseudo-metric s := ρ^{W_{e}} is called the span (pseudo-)metric. This is also normalized, conformally invariant and monotone. By definition, we know s_{R} = 0 if and only if R ∈ O_{AD}.

Since Ω_{e} is a closed subspace of the Hilbert space Ω(R), it has a reproducing kernel, written by K̃_{R}(p, q) = K̃_{R}(z, w)dzdw̄, which is called the exact (or reduced) Bergman

kernel. As before, we can prove that s_{R}(z) = \sqrt{πK̃_{R}(z, z)}. It is clear that s_{R} ≤ b_{R}. The next result is due to Ahlfors and Beurling [1] in the case that R is planar. For the general case, see [9].

Proposition 3.4. We have s_{R} ≤ c_{R} for every Riemann surface R.

Since c_{R} ≤ k_{R}, we also have the estimate s_{R} ≤ k_{R}, which was used effectively in the paper [2] by Beardon and Gehring.

Example 7 (quadratic differentials). Let A(R) be the set of integrable holomorphic quadratic differentials (2-forms) φ = φ(z)dz^{2} on R with norm

∥φ∥_{A(R)} = ∫∫ _{R} |φ| = ∫∫ _{R} |φ(z)|dxdy.

Define the class Q by Q(R) = {φ ∈ A(R); ∥φ∥_{A(R)} ≤ π}. Then the pseudo-metric q := ρ^{Q} is normalized, conformally invariant and monotone. (In this case, m = 2.) In particular, by Theorem 1.2, we see q_{R} ≤ h_{R}. It is essentially well known that q_{R} = 0 if and only if R is conformally equivalent to either ℂ̂, ℂ, ℂ^{∗} = ℂ − {0} or ℂ − {0, 1}. For more details of this pseudo-metric, see [11].

For every element ω ∈ Ω(R), we see ω^{⊗2} = ω(z)^{2}dz^{2} belongs to A(R) and satisfies ∥ω^{⊗2}∥_{A(R)} = ∥ω∥_{Ω(R)}^{2}. Therefore, we have the following:

Proposition 3.5. The inequality b_{R} ≤ q_{R} holds for every Riemann surface R.

The next example may be out of our formulation, however the way to understand logarithmic capacity as in the following is sometimes important.

Example 8 (logarithmic capacity). Let B^{p}(R) be the set of multivalued holomorphic functions f on R such that |f| is single-valued, bounded and satisfies f(p) = 0. Then, the pseudo-metric r defined by

r_{R}(p) = sup{|df|(p); f ∈ B^{p}(R)}

is normalized, conformally invariant and monotone. We should note here that |df| is also single-valued for f ∈ B^{p}(R). We will understand below that r_{R} = 0 if and only if R ∈ O_{G}.

The quantity r_{R}(q) is called the (logarithmic) capacity. In fact, let γ be the Robin constant of R at a point q with respect to a local coordinate α, in other words, Green’s function G_{R}(p, q) of R with pole at q has the local behaviour

G_{R}(p, q) = − log |α(p) − α(q)| + γ + u(p)

near the point q, where u is a harmonic function near q with u(q) = 0. It suffices to see r_{R,α}(w) = e^{−γ}, where w = α(q). Set v(p) = G_{R}(p, q) for p ∈ R. First observe that ω := 2(∂v/∂z)dz is a holomorphic 1-form on R−{q} and ω_{α}(z)+(z −w)^{−1} is holomorphic near w = α(q). We fix a point p_{0} in R other than q. Then the holomorphic function

f(p) = exp (− ∫_{p_{0}}^{p} ω − v(p_{0}))

is multivalued in R but has single-valued modulus and satisfies |f| = e^{−v} < 1. Thus f ∈ B^{q}(R) and we can easily see df_{α}(w) = e^{−γ}.

Let g be an arbitrary multivalued holomorphic function on R with |g| being singlevalued and |g| < 1. Then ψ = − log |g| is positive and superharmonic in R. Since ψ − v is harmonic near the point q or takes the value +∞ at q, we conclude that ψ − v > 0 on R by the minimum principle. This implies |dg|(q) ≤ |df|(q) for such a g. Now we have r_{R,α}(w) = |df|_{R,α}(w) = e^{−γ}.

By this characterization, we have c_{R} ≤ r_{R} for all Riemann surfaces R. We also have the next result due to Burbea [3].

Proposition 3.6. For every Riemann surface R, the inequality r_{R} ≤ k_{R} holds.

Suita [12] conjectured that r_{R} ≤ b_{R} for all Riemann surfaces R. Although this conjecture is not solved yet, recently Ohsawa [7] proved r_{R} ≤ Kb_{R} for all Riemann surfaces R with an absolute constant K ≤ \sqrt{750}. (In [8], the estimate was improved to K ≤ 16\sqrt{2}.)

Summarizing the above results, we have the chain of inequalities

a_{R} ≤ s_{R} ≤ c_{R} ≤ \{ \begin{matrix}r_{R} ≤ k_{R} \\ b_{R} ≤ q_{R} \end{matrix} \} ≤ h_{R}

for every Riemann surface R.

References

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7. T. Ohsawa, Addendum to “On the Bergman kernel of hyperconvex domains”, Nagoya Math. J. 129 (1993), 43–52.

8. , Bergman kernel on Riemann surfaces and the Suita conjecture, RIMS Kōkyūroku 1067 (1998), 89–95, (in Japanese).

9. M. Sakai, On constants in extremal problems of analytic functions, Kōdai Math. Sem. Rep. 21 (1969), 223–225, ibid., 22(1970), 128.

10. L. Sario and K. Oikawa, Capacity Functions, Springer-Verlag, 1969.

11. T. Sugawa, A conformally invariant metric on Riemann surfaces associated with integrable holomorphic quadratic differentials, preprint, 1999.

12. N. Suita, Capacities and kernels on Riemann surfaces, Arch. Rat. Mech. Anal. 46 (1972), 212–217.

13. K. I. Virtanen, Über Extremalfunktionen auf offenen Riemannschen Flächen, Ann. Acad. Fenn. Ser. A. I. 75 (1952), 1–7.

Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan

E-mail address: sugawa@kusm.kyoto-u.ac.jp