A CONFORMALLY INVARIANT METRIC ON RIEMANN SURFACES ASSOCIATED WITH INTEGRABLE HOLOMORPHIC QUADRATIC

DIFFERENTIALS

TOSHIYUKI SUGAWA

Abstract. In this paper, we define a conformally invariant (pseudo-)metric on all Riemann surfaces in terms of integrable holomorphic quadratic differentials and analyze it. This metric is closely related to an extremal problem on the surface. As a result, we have a kind of reproducing formula for integrable quadratic differentials. Furthermore, we establish a new characterization of boundedness of geometry of hyperbolic Riemann surfaces in terms of invariant metrics.

1. Introduction

Throughtout this paper, R denotes a hyperbolic Riemann surface with hyperbolic metric ρ_{R} = ρ_{R}(z)|dz| of constant negative Gaussian curvature −4.

For integers m and n, let ω be a differential (m, n)-form on R, namely, ω is a section of the line bundle K_{R}^{⊗m} ⊗ \overline{K}_{R}^{⊗n}, where K_{R} denotes the canonical bundle over R. For a local coordinate z = α(p) from an open set U_{α} of R onto an open set V_{α} of ℂ, we will write ω = ω_{α}(z)dz^{m}dz̄^{n} on U_{α}, where ω_{α} is a function on V_{α}. Recall that the transition relation ω_{β}(w)(w′)^{m}(\overline{w′})^{n} = ω_{α}(z), where w = β ∘ α^{−1}(z) and w′ = dw/dz. An (m, 0)-form will be just called an m-form. Traditionally, a 2-form is called a quadratic differential. If no confusion will occur, we sometimes omit the suffix indicating the local coordinate.

We also note that in the case m = n = 1/2 the above definition still has meaning because we can well define (w′)^{1/2}(\overline{w′})^{1/2} as |w′| and the positivity of coefficients is preserved by the change of local coordinates. We will write |dz| instead of dz^{1/2}dz̄^{1/2}. A positive (or non-negative) (1/2, 1/2)-form on R will be called a (conformal) metric (or pseudo-metric) on R.

We recall 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 |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).

Date: January 21, 2000.

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

Key words and phrases. quadratic differential, Petersson series, Bergman kernel .

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

We consider the following vector spaces of differentials on R :

Ω(R) = {ω = ω(z)dz; holomorphic 1-form on R, ∥ω∥_{2}^{2} = ∫∫ _{R} |ω(z)|^{2}dxdy < ∞},
A(R) = {φ = φ(z)dz^{2}; holomorphic 2-form on R, ∥φ∥_{1} = ∫∫ _{R} |φ(z)|dxdy < ∞},
B(R) = {φ = φ(z)dz^{2}; holomorphic 2-form on R, ∥φ∥_{∞} = sup_{p∈R} ρ_{R}^{−2}|φ|(p) < ∞}.

The spaces A(R) and B(R) are important in Teichmüller theory and are sometimes called the Bers spaces. A hyperbolic Riemann surface will be called non-exceptional if dim A(R) > 0. Note that an exceptional one is conformally equivalent to a thrice punctured sphere.

We now define two invariant pseudo-metrics b_{R} and q_{R} by using the above spaces. We set

b_{R,α}(z) = sup\{|ω_{α}(z)|; ω ∈ Ω(R) with ∥ω∥_{2} ≤ \sqrt{π}\} and
q_{R,α}(z) = sup{|φ_{α}(z)|^{1/2}; φ ∈ A(R) with ∥φ∥_{1} ≤ π}

for each local coordinate z = α(p) of R.

The first one is known to be expressed by b_{R}(z) = \sqrt{πK_{R}(z, z)} where K_{R}(z, w)dzdw̄ is the Bergman kernel of R. The second one is new as far as the author knows. These pseudo-metrics satisfy the following fundamental properties.

Lemma 1.1 ([20]). The metrics b_{R} and q_{R} are continuous, normalized, conformally invariant and monotonic. More precisely,

1. b_{R} and q_{R} are continuous and log b_{R} and log q_{R} are subharmonic or identically −∞,

2. for the unit disk 𝔻 = {|z| < 1} we have b_{𝔻}(z) = q_{𝔻}(z) = 1/(1 − |z|^{2}),

3. for a biholomorphic map f : R → R′ we have f^{∗}b_{R′} = b_{R} and f^{∗}q_{R′} = q_{R}, and

4. for a subdomain R_{0} of R we have b_{R_{0}} ≥ b_{R} and q_{R_{0}} ≥ q_{R}.

The first property follows from a certain compactness property of closed balls of Ω(R) and A(R). The second property is a consequence of the mean value property of holomorphic functions (see the calculation at the end of Section 2). The other properties can be directly seen.

The next result is one of motivations for the present work.

Proposition 1.2 ([20]). We have

b_{R} ≤ q_{R} ≤ h_{R},

where h_{R} denotes the Hahn metric on R.

The Hahn metric h_{R} = h_{R,α}(z)|dz| is conformally invariant and defined by

h_{R,α}(z) = inf_{f} |(α ∘ f)′(0)|^{−1},

where the infimum is taken over all injective holomorphic maps f from the unit disk 𝔻 into R with f(0) = α^{−1}(z). A systematic treatment of the Hahn metric can be found in [12]. We just recall here a few useful facts.

Lemma 1.3 (Minda [12]). The Hahn metric is comparable with the quasi-hyperbolic metric for a plane domain. Precisely, the following inequality holds for a proper subdomain Ω of ℂ:

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

where δ_{Ω}(z) = min{|z − a|; a ∈ ∂Ω}.

Denote by ι_{R}(p) the injectivity radius of R at the point p. In other words, 2ι_{R}(p) = inf_{γ} ∫_{γ} ρ(z)|dz|, where the infimum is taken over all smooth closed curves γ passing through p which are not contractible in R. Then we have the following result.

Lemma 1.4 (Gotoh [4]).

\frac{coth ι_{R}}{4} ≤ \frac{h_{R}}{ρ_{R}} ≤ coth ι_{R}.

We will need also the following.

Lemma 1.5 (Minda [12]). Let f : R̃ → R be a holomorphic covering map. Then hR̃ ≤ f^{∗}h_{R} holds.

Proof of Proposition 1.2. For each ω ∈ Ω(R) with ∥ω∥_{2} ≤ \sqrt{π}, the square ω⊗ω belongs to A(R) and satisfies ∥ω⊗ω∥_{1} = ∥ω∥_{2}^{2} ≤ π. Therefore, we have |ω(z)| = |ω⊗ω(z)|^{1/2} ≤ q_{R}(z), which implies b_{R}(z) ≤ q_{R}(z). The inequality q_{R} ≤ h_{R} follows from properties 2,3 and 4 in Lemma 1.1 (see [20]).

For a planar Riemann surface R, it is known that b_{R} ≡ 0 if and only if R does not carry the Green function (i.e., R ∈ O_{G}). Furthermore, a puncture is removable with respect to square integrable holormophic 1-forms, and hence we have b_{R−{p_{0}}} ≡ b_{R} for any point p_{0} ∈ R. In contrast, we have the following.

Lemma 1.6. For a non-exceptional hyperbolic Riemann surface R, we have q_{R}(p) > 0 for each point p of R. For a proper subdomain R_{0} of R, the metric b_{R_{0}} is not identically equal to b_{R} on R_{0}.

Proof. When R is planar, R can be embedded in the complex plane ℂ. Then R omits at least three points, say a, b and c, in the plane because R is non-exceptional. Thus the holomorphic quadratic differential dz^{2}/(z − a)(z − b)(z − c) on R is integrable and has no zeros there. When R is not planar, the space Ω(R) has positive dimension. In this case, Virtanen’s theorem (see [18, p. 43]) tells us that b_{R}(p) > 0 at each point p ∈ R. By Proposition 1.2 we have the conclusion in this case, too.

Another motivation comes from the relation between the Bers spaces A(R) and B(R). Once upon a time, it had been conjectured that A(R) ⊂ B(R) for all Riemann surfaces. This was easily proved in the case that R is of finite type. Lehner is the first person who gave a sufficient condition for this inclusion relation to be valid for Riemann surfaces of inifinite topological type (see [7]). Later, Pommerenke [16] disproved the conjecture by showing that there exists a holomorphic 1-form ω on R which is hyperbolically bounded, i.e., ρ_{R}^{−1}|ω| is bounded, but is not square integrable on R for some Riemann surface R.

(Recently, more geometric constructions for counterexamples were given by Ohsawa [15] and Matsuzaki [10].)

Finally, Niebur and Sheingorn [13] gave a complete answer to this problem: A(R) ⊂ B(R) if and only if λ^{∗}(R) > 0, where λ^{∗}(R) denotes the infimum of hyperbolic lengths of those simple closed curves in R which are homotopic to neither a point nor a puncture. We will say that a Riemann surface R with the property λ^{∗}(R) > 0 is of Lehner type. If the infimum λ(R) of hyperbolic lengths of simple closed curves in R which are not homotopic to any point is positive, then R is said to be of bounded geometry. Since 2 inf_{p∈R} ι_{R}(p) = λ(R), this property is equivalent to the condition that the (hyperbolic) injectivity radius has a positive lower bound on R.

Note that λ(R) = 0 if R has a puncture and λ(R) = λ^{∗}(R) otherwise. We have the following characterization of boundedness of geometry in terms of invariant metrics.

Proposition 1.7 (cf. [19]). A hyperbolic Riemann surface R is of bounded geometry, i.e., λ(R) > 0 if and only if the Hahn metric is comparable with the Poincaré metric:

sup_{p∈R} \frac{h_{R}}{ρ_{R}} (p) < +∞.

In this paper, we provide a new characterization of boundedness of geometry as in the following.

Theorem 1.8. A hyperbolic Riemann surface R is of bounded geometry, i.e., λ(R) > 0 if and only if

H(R) := sup_{p∈R} \frac{h_{R}}{q_{R}} (p) < +∞.

Note that there is no order relation between ρ_{R} and q_{R} in general (see the remark made at the end of Section 3). We will give a proof of this in Section 5 with a more concrete estimate (Theorem 5.1).

If A(R) ⊂ B(R), we set

κ(R) = sup{∥φ∥_{∞}; φ ∈ A(R) with ∥φ∥_{1} ≤ π}.

Note that κ(R) < +∞ by the closed graph theorem. If A(R) is not contained in B(R), we set κ(R) = +∞. Recently, Matsuzaki [9] obtained a concrete estimate of this quantity:

\frac{1}{2λ^{∗}(R)} ≤ κ(R) ≤ min \{\frac{c_{1}}{λ^{∗}(R)}, c_{2}\} , (1.1)

where c_{1} and c_{2} are positive absolute constants. The following result will enable us to deduce a refinement of Matsuzaki’s result in the sence that the ratio q_{R}/ρ_{R} may be regarded as a localization of the global quantity κ(R). In fact, as an application, we reproduce Matsuzaki’s inequality with concrete constants (Theorem 4.6) in Section 4.

Proposition 1.9. We have the relation

κ(R) = sup_{p∈R} (\frac{q_{R}}{ρ_{R}} (p))^{2} .

Proof. Setting A_{0}(R) = {φ ∈ A(R); ∥φ∥_{1} ≤ π}, we calculate

κ(R) = sup_{φ∈A_{0}(R)} ∥φ∥_{∞} = sup_{φ∈A_{0}(R)} sup_{p∈R} ρ_{R}^{−2}|φ|(p) = sup_{p∈R} sup_{φ∈A_{0}(R)} ρ_{R}^{−2}|φ|(p) = sup_{p∈R} ρ_{R}^{−2}q_{R}^{2}(p).

In particular, we have the following characterization of Riemann surfaces of Lehner type, which should be compared with Proposition 1.7 and Theorem 1.8.

Corollary 1.10. A hyperbolic Riemann surface R is of Lehner type, i.e., λ^{∗}(R) > 0 if and only if the metric q_{R} is dominated by the Poincaré metric ρ_{R}, that is,

sup_{p∈R} \frac{q_{R}}{ρ_{R}} (p) < +∞.

We now briefly explain the structure of the present paper. In Section 2, we consider an extremal problem in connection with our metric q_{R} to obtain a reproducing formula for A(R). Section 3 will be devoted to a preliminary investigation of q_{R} for doubly connected domains R. Using the results in Section 3, we deduce upper and lower estimates of q_{R} around a short geodesic and a puncture in Section 4. In Section 5, we will prove the characterization theorem (Theorem 1.8) for boundedness of geometry (equivalently, uniform perfectness of the “boundary”) as was promised. We will provide a lower estimate of q_{R} in terms of the (logarithmic) capacity metric for R ∉ O_{G} in Section 6. Section 7 will treat the connection with the usual kernel function for A(R).

2. Extremal problem for quadratic differentials

Which quadratic differential does attain the supremum in the definition of the metric q_{R}? By definition, we see

q_{R,α}(z_{0})^{2} = sup_{φ∈A_{0}(R)} Re φ_{α}(z_{0}), (2.1)

where A_{0}(R) = {φ ∈ A(R); ∥φ∥_{1} ≤ π}. The extremal problem to find a differential φ ∈ A_{0}(R) attaining the supremum in (2.1) has a unique solution and leads to a kind of reproducing formula for integrable holomorphic quadratic differentials on R.

Theorem 2.1. Let z = α(p) be a local coordinate of a non-exceptional Riemann surface R. For each point z_{0} ∈ V_{α} there exists a unique φ ∈ A(R) such that

q_{R,α}(z_{0})^{2} = φ_{α}(z_{0}).

Furthermore, ∥φ∥_{1} = π and the following formula holds:

ψ_{α}(z_{0}) = \frac{q_{R,α}(z)^{2}}{π} ∫∫ _{R} \frac{|φ|}{φ}ψ (ψ ∈ A(R)). (2.2)

Conversely, if a non-zero element φ in A(R) satisfies relation (2.2), then πφ/∥φ∥_{1} is a solution of the extremal problem (2.1).

Proof. We omit the suffix α below. Take a sequence φ_{n} in A_{0}(R) such that Re φ_{n}(z_{0}) → q_{R}(z_{0})^{2} as n → ∞. Since A_{0}(R) is a normal family, we may assume that the sequence φ_{n} converges to a holomorphic quadratic differential φ uniformly on each compact subset of R. Recalling Lemma 1.6, we can see

q_{R}(z_{0})^{2} = Re φ(z_{0}) = φ(z_{0}) > 0. (2.3)

Fatou’s lemma yields

∥φ∥_{1} ≤ \underline{lim}_{n→∞} ∥φ_{n}∥_{1} ≤ π.

In fact, ∥φ∥_{1} = π because of the extremality of φ. Thus we have shown the existence of an extremal differential φ.

Next we prove equality (2.2). Let ψ ∈ A(R) be linearly independent of φ over the real field and consider the function f(t) = ∥φ + tψ∥_{1} of t ∈ ℝ. Then the extremality of φ implies the inequality

\frac{Re (φ(z_{0}) + tψ(z_{0}))}{f(t)} ≤ \frac{Re φ(z_{0})}{f(0)} = \frac{q_{R}(z_{0})^{2}}{π} .

Since ||φ + tψ| − |φ|| ≤ |t||ψ| and since |a + tb| = |a| + t|a|Re (b/a) + o(t) as t → 0 in ℝ for complex numbers a and b, Lebesgue’s dominated convergence theorem produces

f(t) = f(0) + t Re ∫∫ _{R} \frac{|φ|}{φ}ψ + o(t)

as t → 0. Hence we have

q_{R}(z_{0})^{2} + t Re ψ(z_{0}) ≤ \frac{q_{R}(z_{0})^{2}}{π} (π + t Re ∫∫ _{R} \frac{|φ|}{φ}ψ + o(t))

as t → 0. Since t may take both signs, this inequality forces

Re ψ(z_{0}) = π^{−1}q_{R}(z_{0})^{2}Re ∫∫ _{R} |φ|ψ/φ.

This equality also holds for e^{iθ}ψ instead of ψ, so we can eliminate the symbol “Re” from this equality. Thus (2.2) is now established.

Finally, we show the uniqueness of the extremal differential φ. Suppose that φ_{1} ∈ A_{0}(R) is also an extremal differential at z_{0}, i.e., q_{R}(z_{0})^{2} = φ_{1}(z_{0}). Then, applying ψ = φ_{1} to (2.2), we obtain

q_{R}(z_{0})^{2} = φ_{1}(z_{0}) = \frac{q_{R}(z_{0})^{2}}{π} Re ∫∫ _{R} \frac{|φ|}{φ}φ_{1}.

We now estimate

π = Re ∫∫ _{R} \frac{|φ|}{φ}φ_{1} ≤ |∫∫ _{R} \frac{|φ|}{φ}φ_{1}| ≤ ∫∫ _{R} |φ_{1}| ≤ π.

Therefore, equalities hold above and the argument of |φ|φ_{1}/φ must be identically 0, which implies that φ_{1}/φ is a positive constant. Thus φ_{1} = φ and the uniqueness is now proved. The latter part of the theorem is trivial.

Remark. The extremal differential φ above depends on the choice of the local coordinate α. To be more precise, we write φ = φ_{R}^{α,z_{0}}. If we take another local coordinate β around p_{0} = α^{−1}(z_{0}), then we have

φ_{R}^{β,w(z_{0})} \frac{\overline{w′(z_{0})}}{w′(z_{0})} = φ_{R}^{α,z_{0}},

where w(z) = β ∘ α^{−1}(z). In other words, the extremal differential at p can be written as

φ_{R}(r, p) = φ_{R,α,β}(ζ, z)dζ^{2} \frac{dz̄}{dz},

where ζ = α(r) , z = β(p) and φ_{R,α,β}(ζ, z) = φ_{R,α}^{β,z}(ζ). This means that φ_{R} is a section of the line bundle K_{R}^{⊗2} ⊠ (K_{R}^{∗} ⊗ \overline{K}_{R}) over R × R, where ⊠ stands for the external tensor product.

It is convenient to record the following translation formula under the notation in the above remark.

Lemma 2.2. Let D and D′ be plane domains and F : D → D′ be a conformal map. Then we have

φ_{D′}(F (ζ), F (z))F′(ζ)^{2} \frac{\overline{F′(z)}}{F′(z)} = φ_{D}(ζ, z).

In the case that R is the unit disk 𝔻 with the canonical coordinate α(z) = z, from the mean value property of integrable holomorphic functions on 𝔻 :

f(0) = \frac{1}{π} ∫∫ _{𝔻} f(z)dxdy,

it follows that φ ≡ 1 is the extremal quadratic differential at the origin. By Lemma 2.2, we see that

φ_{𝔻}(ζ, z) = \frac{(1 − |z|^{2})^{2}}{(1 − z̄ζ)^{4}} .

Therefore, we have a reproducing formula for A(𝔻) by (2.2):

ψ(z) = \frac{1}{π(1 − |z|^{2})^{2}} ∫∫ _{𝔻} ψ(ζ) (\frac{1 − z̄ζ}{1 − zζ̄})^{2} dξdη.

3. Estimate on annulus

In this section, we deduce an estimate of q_{𝔸}(w) for a round annulus 𝔸 = 𝔸_{r} = {w ∈ ℂ; r < |w| < 1/r} for 0 < r < 1.

Theorem 3.1. For the annulus 𝔸 = {w; r < |w| < 1/r}, we have the estimate

Q_{1}(w) ≤ q_{𝔸}(w)^{2} ≤ Q_{2}(w), r < |w| < 1/r,

where

Q_{1}(w) = max \{(\frac{r}{1 − (r|w|)^{2}} )^{2} , (\frac{r}{|w|^{2} − r^{2}} )^{2} , \frac{1}{4|w|^{2} log 1/r}, \frac{r}{2|w|(1 − r^{2})}, \frac{r}{2|w|^{3}(1 − r^{2})}\}

and Q_{2} is determined by the relations Q_{2}(1/w)|w|^{−4} = Q_{2}(w) and

Q_{2}(w) = max \{\frac{1}{2|w|^{2}(log \frac{1}{r|w|} + r|w| − 1)}, \frac{1}{2|w|^{2}(log \frac{r}{|w|} + \frac{|w|}{r} − 1)}\}

for |w| ≤ 1. We also have Q_{2}(w) ≤ 9Q_{1}(w) for w ∈ 𝔸.

Before proceeding to the proof, we make several observations. Recall that the pullback j^{∗} by the automorphism j(z) = 1/z of 𝔸 is defined by j^{∗}P (w) = P (1/w)|w|^{−4} for a (1, 1)- form P = P (w)|dw|^{2} on 𝔸. We denote by P_{1}, P_{2}, P_{3}, P_{4} and P_{5} the five functions in the above definition of Q_{1} in the order of appearance, which are all regarded as (1, 1)-forms on 𝔸. Note that j^{∗}P_{1} = P_{2}, j^{∗}P_{3} = P_{3} and j^{∗}P_{4} = P_{5}. In particular, we know that Q_{1}, Q_{2} and q_{𝔸}^{2} are all invariant under the pullback by the analytic involution j. Setting m(t) = − log t + t − 1 for t > 0, we have the expression

Q_{2}(w) = \begin{cases}1/2|w|^{2}m(|w|/r) & if r < |w| ≤ r_{0} \\ 1/2|w|^{2}m(r|w|) & if r_{0} ≤ |w| ≤ 1 \\ 1/2|w|^{2}m(r/|w|) & if 1 ≤ |w| ≤ 1/r_{0} \\ 1/2|w|^{2}m(1/r|w|) & if 1/r_{0} ≤ |w| < 1/r, \end{cases}

where r_{0} = 2(log 1/r)/(1/r − r). Note that r_{0} = t/ sinh t if we set r = e^{−t} and that r < r_{0} < 1. We also remark that |w| ≤ r_{0} if and only if P_{3}(w) ≤ P_{5}(w). Observe that the function m(t) is convex and takes its minimum at t = 1. For fixed t_{0} ∈ (0, 1) and t_{1} ∈ (1, +∞), the following estimates will be useful later:

(3.1)  m(t) ≥ \frac{m(t_{0})}{log 1/t_{0}} log \frac{1}{t} for 0 < t ≤ t_{0},
(3.2)  m(t) ≥ \frac{(1 − t)^{2}}{2} for t_{0} ≤ t ≤ 1,
(3.3)  m(t) ≥ \frac{(t − 1)^{2}}{2t_{1}} for 1 ≤ t ≤ t_{1}, and
(3.4)  m(t) ≥ \frac{m(t_{1})}{t_{1}}t for t_{1} ≤ t.

Taking w = 1, we have the following

Corollary 3.2. For 𝔸 = {r < |w| < 1/r}, we have

max \{(\frac{r}{1 − r^{2}} )^{2} , \frac{1}{4 log 1/r}\} ≤ q_{𝔸}(1)^{2} ≤ \frac{1}{2(− log r + r − 1)}.

Proof of Theorem 3.1. First, we show the inequality Q_{1} ≤ q_{𝔸}^{2}. Since 𝔻_{1/r} = {|z| < 1/r} ⊃ 𝔸, the monotonicity of the metric q implies ρ_{𝔻_{1/r}} = q_{𝔻_{1/r}} ≤ q_{𝔸} in 𝔸. Namely, P_{1}^{1/2} = r/(1 − r^{2}|w|^{2}) ≤ q_{𝔸}. We next consider the differential φ_{j}(w)dw^{2} = w^{j}dw^{2} on 𝔸 for an integer j. By definition, we have π|φ_{j}(w)|/∥φ_{j}∥_{1} ≤ q_{𝔸}(w)^{2}. An easy calculation shows that ∥φ_{−2}∥_{1} = 4π log 1/r and ∥φ_{−1}∥_{1} = 2π(1/r − r). Thus we have P_{3} ≤ q_{𝔸}^{2} and P_{4} ≤ q_{𝔸}^{2}. The other inequalities P_{2} ≤ q_{𝔸}^{2} and P_{5} ≤ q_{𝔸}^{2} follow from the invariance observed above.

Secondly, we show the inequality q_{𝔸}^{2} ≤ Q_{2}. Let φ(w)dw^{2} be any element of A(𝔸). Take w ∈ 𝔸 and fix it. By the symmetry, we may assume |w| ≤ 1. Let s and t be any numbers such that r < s < |w| < t < 1/r. Cauchy’s integral formula then implies

w^{2}φ(w) = \frac{1}{2πi} ∫_{|ζ|=t} \frac{ζ^{2}φ(ζ)}{ζ − w}dζ − \frac{1}{2πi} ∫_{|ζ|=s} \frac{ζ^{2}φ(ζ)}{ζ − w}dζ = ψ_{0}(w) + ψ_{1}(w).

We calculate

|ψ_{0}(w)| ≤ \frac{1}{2π} ∫_{0}^{2π} \frac{t^{2}|φ(te^{iθ})|}{t − |w|}tdθ.

Integrating both sides of |ψ_{0}(w)|(t − |w|)/t^{2} ≤ ∫_{0}^{2π} |φ(te^{iθ})|tdθ with respect to t from |w| to 1/r, we obtain

|ψ_{0}(w)|m(r|w|) ≤ \frac{1}{2π} ∫_{|w|}^{1/r} ∫_{0}^{2π} |φ(te^{iθ})|tdθdt = \frac{1}{2π} ∫∫ _{|w|<|ζ|<1/r} |φ(ζ)|dξdη.

Similarly, we have

|ψ_{1}(w)|m(|w|/r) ≤ \frac{1}{2π} ∫∫ _{r<|ζ|<|w|} |φ(ζ)|dξdη.

Combining these two inequalities, we see

|w^{2}φ(w)| ≤ |ψ_{0}(w)| + |ψ_{1}(w)| ≤ \frac{1}{2π} max \{\frac{1}{m(r|w|)}, \frac{1}{m(|w|/r)}\} ∥φ∥_{1}.

The last inequality yields |w|^{2}q_{𝔸}(w)^{2} ≤ \frac{1}{2} max\{1/m(r|w|), 1/m(|w|/r)\}, which is nothing but the desired one.

Finally, we show Q_{2}(w) ≤ 9Q_{1}(w). We assume |w| ≤ 1 as well. First we show 1/2|w|^{2}m(r|w|) ≤ 9Q_{1}(w). When r|w| ≥ 1/2, by (3.2) we have

\frac{1}{2|w|^{2}m(r|w|)P_{1}(w)} ≤ \frac{(1 − r^{2}|w|^{2})^{2}}{r^{2}|w|^{2}(1 − r|w|)^{2}} = (1 + \frac{1}{r|w|})^{2} ≤ 9.

When r|w| ≤ 1/2, by (3.1) we have

\frac{1}{2|w|^{2}m(r|w|)P_{3}(w)} = \frac{2 log 1/r}{m(r|w|)} ≤ \frac{2 log 2 · log 1/r}{m(1/2) log 1/r|w|} ≤ \frac{2 log 2}{m(1/2)} = 7.17 ⋯ < 9.

So we have shown 1/2|w|^{2}m(r|w|) ≤ 9Q_{1}(w) for r < |w| ≤ 1.

Next we prove 1/2|w|^{2}m(|w|/r) ≤ 9Q_{1}(w). When |w|/r ≥ 2, by (3.4) we obtain

\frac{1}{2|w|^{2}m(|w|/r)P_{5}(w)} ≤ \frac{2|w|^{3}(1 − r^{2})}{r} · \frac{2}{2|w|^{2}m(2)|w|/r} ≤ \frac{2}{m(2)} = 6.51 ⋯ < 9.

On the other hand, when |w|/r ≤ 2, by (3.3), we have

\frac{1}{2|w|^{2}m(|w|/r)P_{2}(w)} ≤ \frac{2(|w|^{2} − r^{2})^{2}}{r^{2}|w|^{2}(|w|/r − 1)^{2}} = 2 (1 + \frac{r}{|w|})^{2} < 8 < 9.

Hence, we see 1/2|w|^{2}m(|w|/r) ≤ 9Q_{1}(w) for r < |w| ≤ 1. Now the proof is complete.

We have also an estimate on the punctured disk 𝔻^{∗} = 𝔻−{0} in a similar way as above. Note that the function wφ(w) is always holomorphic in the unit disk for each φ ∈ A(𝔻^{∗}).

Theorem 3.3. We have Q_{1}(w) ≤ q_{𝔻^{∗}}(w)^{2} ≤ Q_{2}(w) for w ∈ 𝔻^{∗}, where

Q_{1}(w) = max \{\frac{1}{(1 − |w|^{2})^{2}} , \frac{1}{2|w|}\} and Q_{2}(w) = \frac{1}{2|w|^{2}(|w|^{−1} − 1 − log |w|^{−1})}.

In particular, q_{𝔻^{∗}}(w)^{2} = \frac{1+o(1)}{2|w|} as w → 0. Furthermore, Q_{2}(w) ≤ 8Q_{1}(w) holds for w ∈ 𝔻^{∗}.

Corollary 3.4. The metric q_{R} is incomplete at a puncture of R.

Proof. Suppose that ι : 𝔻^{∗} → R is an injective holomorphic map such that the induced map ι_{∗} : π_{1}(𝔻^{∗}, ∗) → π_{1}(R, ∗) is injective. Therefore, the puncture of 𝔻^{∗} corresponds to a puncture of R. Let γ_{0} : (0, 1/2] → 𝔻^{∗} be the curve defined by γ_{0}(t) = t. Then γ = ι ∘ γ_{0} is a proper map from (0, 1/2] into R.

The monotonicity of q (Lemma 1.1) implies ι^{∗}q_{R} ≤ q_{𝔻^{∗}}. Thus we have

∫_{γ} q_{R} = ∫_{γ_{0}} ι^{∗}q_{R} ≤ ∫_{γ_{0}} q_{𝔻^{∗}}(z)|dz| ≤ const. ∫_{0}^{1/2} t^{−1/2}dt < ∞.

Remark. From the estimates in this section, we know that there is no natural order relation between q_{R} and ρ_{R}. Actually, by Corollary 3.2, we see that

\frac{q_{𝔸_{r}}(1)^{2}}{ρ_{𝔸_{r}}(1)^{2}} ∼ \frac{(log 1/r)^{2}}{r − 1 + log 1/r} ∼ log \frac{1}{r}

as r → 0 + . On the other hand, since ρ_{𝔻^{∗}}(w) = 1/2|w| log(1/|w|), we see that

\frac{q_{𝔻^{∗}}(w)^{2}}{ρ_{𝔻^{∗}}(w)^{2}} ∼ |w|(log 1/|w|)^{2} = o(1)

as w → 0.

4. Estimate on a collar

As is well recognized, the presence of short geodesics has a significant effect on the geometry of the unit ball of the Banach space A(R) (see, for example, McMullen [11] or Ohsawa [14]). In this section, we estimate the magnitude of q_{R} on the collar of a short geodesic in R by utilizing the analysis made in the preceeding section.

Let γ be a simple closed geodesic of R with hyperbolic length ℓ. We now take a holomorphic universal covering u : ℍ → R of R with the covering transformation g_{0}(z) = e^{2ℓ}z covering the geodesic γ, where ℍ deontes the upper half plane {z ∈ ℂ; Im z > 0}. We denote by G the covering transformation group of u and by G_{0} the subgroup of G generated by g_{0}. Note that G is a torsion-free discrete subgroup of Aut (ℍ) = PSL(2, ℝ). The angular region ℍ_{θ} = {z ∈ ℍ; | arg(z/i)| < θ} for 0 < θ ≤ π/2 is invariant under the action of G_{0}. The covering map E_{ℓ}(z) = exp(πiℓ^{−1} log(z/i)) maps ℍ_{θ} onto the annulus 𝔸(θ) = {exp(−πθ/ℓ) < |w| < exp(πθ/ℓ)} and corresponds to the quotient map ℍ_{θ} → ℍ_{θ}/G_{0}. For abbreviation, we write r = exp(−π^{2}/2ℓ) and 𝔸 = 𝔸(π/2) = {r < |w| < 1/r}. The induced covering f_{γ} : 𝔸 → R determined by u = f_{γ} ∘ E_{ℓ} is called the annular covering of R associated with γ.

If ℍ_{θ} is precisely invariant under G_{0} in G, equivalently, if the map f_{γ} is injective in 𝔸(θ), the image C_{γ}(θ) = f_{γ}(𝔸(θ)) is called the collar of γ with angle θ. For the collar C_{γ}(θ), the map α_{γ} := f_{γ}^{−1} : C_{γ}(θ) → 𝔸(θ) is served as a local coordinate of R, which will be called the canonical coordinate for γ. The collar lemma is an important tool for our aim. The proof can be found, for example, in [1].

Lemma 4.1 (The collar lemma). A simple closed geodesic of length ℓ has a collar of angle θ, where θ is determined by tan θ sinh ℓ = 1.

Since arctan(sinh(x)) is concave, the function arctan(sinh(x))/x is monotonically decreasing in x > 0. In particular, we have 1 > arctan(sinh(x))/x, and hence π/2 − θ = arctan(sinh(ℓ)) < ℓ, where θ is the angle in the statement of Lemma 4.1. Therefore, 𝔸(θ) ⊃ 𝔸_{1} := 𝔸(θ_{1}) = {e^{π}r < |w| < 1/e^{π}r}, where θ_{1} = π/2 − ℓ. We now have the following weak version of the collar lemma.

Corollary 4.2. Let f_{γ} : 𝔸 = {r < |w| < 1/r} → R be the annular covering of R associated with a simple closed geodesic γ with length ℓ < π/2, where r = exp(−π^{2}/2ℓ). Then f_{γ} is injective in 𝔸_{1} = {e^{π}r < |w| < 1/e^{π}r}.

Set 𝔸_{2} = 𝔸(θ_{2}) = {e^{2π}r < |w| < 1/e^{2π}r} for a simple closed geodesic γ with length ℓ < π/4, where θ_{2} = π/2−2ℓ. The next result will be used to control quadratic differentials on the collar C_{γ}(θ_{2}).

Lemma 4.3. Let γ be a simple closed geodesic with length ℓ ≤ π/8 = 0.3926 … in R. Then

q_{𝔸_{1}}(w)^{2} ≤ \frac{3}{16π|w|^{2}} (w ∈ 𝔸_{2}).

Proof. By the symmetry, we may assume e^{2π}r ≤ |w| ≤ 1, where r = exp(−π^{2}/2ℓ). Under the notation in Theorem 3.1, we obtain 2|w|^{2}q_{𝔸_{1}}(w)^{2} ≤ 2|w|^{2}Q_{2}(w) = 1/ min{m(e^{π}r|w|), m(|w|/e^{π}r)}. Noting e^{π}r|w| ≤ e^{π}r < 1 and 1 < e^{π} ≤ |w|/e^{π}r, we have

(4.1) m(e^{π}r|w|) ≥ m(e^{π}r) > \frac{π^{2}}{2ℓ} − π − 1 ≥ \frac{π^{2}}{2ℓ} − \frac{(π + 1)π}{8ℓ} = \frac{π(3π − 1)}{8ℓ} > \frac{π^{2}}{3ℓ} ≥ \frac{8π}{3}

and m(|w|/e^{π}r) ≥ m(e^{π}) = 18.999 ⋯ > 8π/3. Hence we see |w|^{2}q_{𝔸_{1}}(w)^{2} > 3/16π|w|^{2}.

We also need a “puncture version” of the collar lemma, which is a consequence of the Shimizu-Leutbecher lemma (see, for example, [8, II.C.5]). Let γ be a simple short loop around a puncture in R. We take a holomorphic universal covering u : 𝔻 → R such that the covering transformation group G of u contains the parabolic element g_{0}(z) = z + 1 which covers γ. Note that the domain ℍ(k) = {z ∈ ℍ; Im z > k} is invariant under g_{0} for k ≥ 0. Set 𝔻^{∗}(k) = E_{0}(ℍ(k)) = {0 < |w| < e^{−2πk}}, where E_{0}(z) = exp(2πiz). Moreover we set 𝔻_{1}^{∗} = 𝔻^{∗}(1) and 𝔻_{2}^{∗} = 𝔻^{∗}(\sqrt{2}). We denote by f_{γ} : 𝔻^{∗} = 𝔻^{∗}(0) → R the annular covering associated with γ, namely, u = f_{γ} ∘ E_{0}.

Lemma 4.4. Let γ be a simple loop corresponding to a puncture of R. Then the annular covering f_{γ} : 𝔻^{∗} → R is injective in 𝔻_{1}^{∗}.

The biholomorphic map α_{γ} = (f_{γ})^{−1} : f_{γ}(𝔻_{1}^{∗}) → 𝔻_{1}^{∗} will be called the canonical coordinate of R around the puncture corresponding to γ.

The next lemma is a variant of “thick-thin decomposition”.

Lemma 4.5. Let ε_{0} = \frac{1}{4} log 2 = 0.17328 …. If a point p in R has injectivity radius ι_{R}(p) < ε_{0}, then p is contained in either f_{γ}(𝔸_{2}) for a simple closed geodesic γ of length less than π/8 or f_{γ}(𝔻_{2}^{∗}) for a simple loop γ corresponding to a puncture.

Proof. Let γ_{1} be a non-contractible closed curve passing through p which minimizes the hyperbolic length, i.e., ∫_{γ_{1}} ρ_{R} = 2ι_{R}(p). It is easily verified that γ_{1} is simple. First we consider the case that γ_{1} has a freely homotopic simple closed geodesic γ of length ℓ. By assumption, we note that ℓ ≤ 2ι_{R}(p) < 2ε_{0} < π/8. The hyperbolic distance d between ite^{iθ}

and ite^{iθ+2ℓ}, where t > 0, satisfies tanh d = 1/\sqrt{1 + cos^{2} θ/ sinh^{2} ℓ}. If π/2−2ℓ ≤ |θ| ≤ π/2 we have cos θ ≤ sin(2ℓ). So, we have cos θ/ sinh ℓ ≤ sin(2ℓ)/ sinh ℓ < 2ℓ/ℓ = 2. Thus d > arctanh(1/\sqrt{5}) > arctanh(1/3) = 2ε_{0}, which means that if a closed curve passing through a point in the outside of f_{γ}(𝔸_{2}) = f_{γ}(𝔸(π/2 − 2ℓ)) is freely homotopic to γ, then its length is necessarily at least 2ε_{0}. Hence p ∈ f_{γ}(𝔸_{2}) in this case.

Since the hyperbolic distance d between ik and ik + 1, where 0 < k ≤ \sqrt{2} satisfies d = arctanh(1/\sqrt{1 + 4k^{2}}) ≥ arctanh(1/3) = 2ε_{0}, the case that γ_{1} corresponds to a puncture can be handled in the same manner as above.

As an application of our metric q_{R}, now we are ready to prove the following theorem, which is a more concrete form of Matsuzaki’s result. Note that we can read more detailed information on the relationship between A(R) and B(R) from our proof presented below. Recall also the fact that 1/2λ^{∗}(R) ≤ κ(R) from (1.1).

Theorem 4.6.

κ(R) ≤ max \{\frac{6}{λ^{∗}(R)}, 34\} .

Proof. By Proposition 1.9, we have only to show that (q_{R}/ρ_{R})^{2} ≤ max{6/λ^{∗}(R), 34}. Consider the function F = q_{R}/ρ_{R} on R. Let p ∈ R be arbitrary. If ι_{R}(p) ≥ ε_{0} = arctanh(1/3)/2, by Proposition 1.2 and Lemma 1.4, we have F (p) ≤ (h_{R}/ρ_{R})(p) ≤ coth ι_{R}(p) ≤ coth ε_{0} = 3 + 2\sqrt{2}. Note here that (3 + 2\sqrt{2})^{2} = 33.970 ⋯ < 34.

Otherwise, by virtue of Lemma 4.5, we have the following two cases:

• Case 1: p ∈ f_{γ}(𝔸_{2}) for a simple closed geodesic γ with length ℓ < π/8.

Let w be a point in 𝔸_{2} with f_{γ}(w) = p. We may assume |w| ≤ 1 and write log |w| = −πθ/ℓ, where 0 ≤ θ ≤ π/2 − 2ℓ. Note that (f_{γ})^{∗}ρ_{R}(w) = ρ_{𝔸}(w) = ℓ/2π|w| cos θ. By Theorem 3.1, we obtain

F (p)^{2} ≤ \frac{q_{𝔸_{1}}(w)^{2}}{ρ_{𝔸}(w)^{2}} ≤ \frac{2π^{2}}{ℓ^{2}} max \{\frac{cos^{2} θ}{m(e^{π}r|w|)}, \frac{cos^{2} θ}{m(|w|/e^{π}r)}\} .

First, we see cos^{2} θ/m(e^{π}r|w|) ≤ 1/m(e^{π}r) < 3ℓ/π^{2} from (4.1). Second, by (3.4), m(|w|/e^{π}r) ≥ m(e^{π})|w|/e^{2π}r holds, and hence

\frac{cos^{2} θ}{m(|w|/e^{π}r)} ≤ \frac{e^{2π}r}{m(e^{π})}e^{πθ/ℓ} cos^{2} θ.

Since the function e^{πx/2ℓ} cos x is monotonically increasing in 0 ≤ x ≤ π/2 − 2ℓ, we have cos^{2} θ/m(|w|/e^{π}r) ≤ sin^{2}(2ℓ)/m(e^{π}) ≤ 4ℓ^{2}/m(e^{π}). Hence, we have

F (p)^{2} ≤ \frac{2π^{2}}{ℓ^{2}} max \{\frac{3ℓ}{π^{2}} , \frac{4ℓ^{2}}{m(e^{π})}\} = max \{\frac{6}{ℓ}, \frac{8π^{2}}{m(e^{π})}\} .

We note here 8π^{2}/m(e^{π}) = 4.1558 ….

• Case 2: p ∈ f_{γ}(𝔻_{2}^{∗}) for a simple closed curve γ around a puncture.

Let w be a point in 𝔻_{2}^{∗} with f_{γ}(w) = p. Note that 0 < |w| < e^{−2\sqrt{2}π} and that f_{γ}^{∗}ρ_{R}(w) = ρ_{𝔻^{∗}}(w) = 1/2|w| log(1/|w|). We now use Theorem 3.3 to show

F (p)^{2} ≤ \frac{q_{𝔻_{1}^{∗}}(w)^{2}}{ρ_{𝔻^{∗}}(w)^{2}} = (2|w| log |w|)^{2}e^{4π}q_{𝔻^{∗}}(e^{2π}w) ≤ \frac{(2|w| log |w|)^{2}}{2|w|^{2}m(1/e^{2π}|w|)} = \frac{2(log |w|)^{2}}{m(1/e^{2π}|w|)}.

The inequality m(1/e^{2π}|w|) ≥ e^{−2\sqrt{2}π}m(e^{2(\sqrt{2}−1)π})/|w| follows from (3.4), and then

F (p)^{2} ≤ \frac{2e^{2\sqrt{2}π}|w|(log |w|)^{2}}{m(e^{2(\sqrt{2}−1)π)})} .

We can easily check that the function x(log x)^{2} is monotonically increasing in 0 < x ≤ e^{−2\sqrt{2}π}(< e^{−2}), so we have

F (p)^{2} ≤ \frac{16π^{2}}{m(e^{2(\sqrt{2}−1)π})} = 15.957 ….

Summing up these estimates, we obtain the desired inequality.

We next concentrate on the lower estimate of q_{R} near γ. To this end, we use the Petersson series for γ. An estimate similar to ours was given by Wolpert [21]. Another approach appealing to Ahlfors’ trick can be found in [10]. We provide an estimate independently for the sake of explicitness.

We denote by A(ℍ, G) the Banach space of those holomorphic automorphic 2-forms φ = φ(z)dz^{2} on ℍ for a torsion-free Fuchsian group G which are integrable on ℍ/G, namely, φ(g(z))g′(z)^{2} = φ(z) for all g ∈ G and ∥φ∥_{A(ℍ,G)} = ∫∫_{ω} |φ(z)|dxdy < +∞, where ω is a fundamental domain for G with Area(∂ω) = 0. Note that A(ℍ, G) is naturally isomorphic to the Banach space A(ℍ/G) via the pullback by the canonical projection ℍ → ℍ/G.

Let γ be a simple closed curve of length ℓ on R. Choose a Fuchsian group G uniformizing R in such a way that the element g_{0}(z) = e^{2ℓ}z in G covers γ. Under the above notation and normalization, the Petersson series φ_{γ} ∈ A(R) for γ is defined by

f^{∗}φ_{γ} = Θ_{G_{0}\G}φ_{0},

where φ_{0} = dz^{2}/z^{2} and Θ_{G_{0}\G} denotes the relative Poincaré series operator for G_{0}\G (see [6] for details), namely, for φ ∈ A(ℍ, G_{0})

Θ_{G_{0}\G}φ = ∑_{g∈G_{0}\G}(φ ∘ g)(g′)^{2},

where the summation is taken over all representatives of right cosets of G_{0} in G. Note that ∥Θ_{G_{0}\G}φ∥_{A(ℍ,G)} ≤ ∥φ∥_{A(ℍ,G_{0})}. Since ∥φ_{0}∥_{A(ℍ,G_{0})} = 2πℓ, we have ∥φ_{γ}∥_{A(R)} ≤ 2πℓ.

Let μ_{0} be the (1, 1)-form |φ_{0}| and set μ = Θ_{G_{0}\G}μ_{0} = ∑_{g∈G_{0}\G} g^{∗}μ_{0} = μ_{0} + μ_{1}, where g^{∗}μ_{0} = (μ_{0} ∘ g)|g′|^{2}. Let ω be a fundamental domain for G such that ℍ_{θ_{1}} ∩ W ⊂ ω, where W = {z ∈ ℍ; e^{−ℓ} < |z| < e^{ℓ}}. Noting ∫∫_{ω} g^{∗}μ_{0} = ∫∫_{g(ω)} μ_{0} and (g_{0})^{∗}μ_{0} = μ_{0}, we calculate ∫∫_{ω} μ = ∫∫_{W} μ_{0} = 2πℓ. On the other hand, we have ∫∫_{ω} μ_{0} ≥ ∫∫_{ℍ_{θ_{1}}∩W} μ_{0} = 4θ_{1}ℓ. Hence, ∫∫_{ω} μ_{1} ≤ 2(π − 2θ_{1})ℓ = 4ℓ^{2}. Set ψ_{0} = −(ℓ/π)^{2}dw^{2}/w^{2}, then we directly see E_{ℓ}^{∗}ψ_{0} = φ_{0}. If we write f_{γ}^{∗}φ_{γ} = ψ_{0} + ψ_{1}, then we have ∥ψ_{1}∥_{A(𝔸_{1})} ≤ ∫∫_{ℍ_{θ_{1}}∩W} μ_{1} ≤ 4ℓ^{2}.

Now we assume that ℓ is sufficiently small, for example, say ℓ ≤ π/8. By Lemma 4.3, we obtain

|ψ_{1}(w)| ≤ \frac{q_{𝔸_{1}}(w)^{2}}{π} ∥ψ_{1}∥_{A(𝔸_{1})} ≤ \frac{3}{16π^{2}|w|^{2}} 4ℓ^{2} = \frac{3ℓ^{2}}{4π^{2}|w|^{2}}

for w ∈ 𝔸_{2}. Thus,

|f_{γ}^{∗}φ_{γ}(w)| ≥ |ψ_{0}(w)| − |ψ_{1}(w)| ≥ \frac{ℓ^{2}}{4π^{2}|w|^{2}}

for w ∈ 𝔸_{2}. Recalling the fact ∥φ_{γ}∥_{A(R)} ≤ 2πℓ, we obtain q_{R,α_{γ}}(w)^{2} = f_{γ}^{∗}q_{R}(w)^{2} ≥ π|f_{γ}^{∗}φ_{γ}(w)|/∥φ_{γ}∥_{A(R)} ≥ ℓ/8π^{2}|w|^{2}. Thus we have shown the following estimate, which may be served for the future use.

Proposition 4.7. For a simple closed geodesic γ with length ℓ ≤ π/8, we have f_{γ}^{∗}q_{R}(w)^{2} ≥ ℓ/8π^{2}|w|^{2} for w ∈ 𝔸_{2}.

5. A characterization of boundedness of geometry

We can now obtain the following estimate, from which we can deduce Theorem 1.8.

Theorem 5.1. Let R be a non-exceptional hyperbolic Riemann surface and denote by H(R) the supremum of the function h_{R}/q_{R} on R. Then, there exists an absolute constant C such that

\frac{1}{λ(R)} ≤ CH(R)^{2}.

On the other hand, if λ(R) > 0 we have

H(R) ≤ K(λ(R)) < +∞,

where K(λ(R)) is a positive constant depending only on λ(R).

Proof. First, we prove the first inequality. Suppose that R has a puncture and let γ be a simple loop around the puncture. By Lemma 4.4 and Theorem 3.3, we then have f_{γ}^{∗}q_{R}(w)^{2} ≤ q_{𝔻_{1}^{∗}}(w)^{2} ∼ 1/|w| as w → 0. On the other hand, from Lemmas 1.3 and 1.5, we can deduce f_{γ}^{∗}h_{R}(w) ≥ h_{𝔻^{∗}}(w) ∼ 1/|w| as w → 0 Thus we have (q_{R}/h_{R})(f_{γ}(w))^{2} = O(|w|) as w → 0, which implies H(R) = +∞. Hence, we conclude that R has no punctures if H(R) < +∞.

Next, let γ be a simple closed geodesic with length ℓ ≤ π/8. Then, by Corollaries 4.2 and 3.2, f_{γ}^{∗}q_{R}(1) ≤ q_{𝔸_{1}}(1) ≤ 1/2m(e^{π}r). Using (4.1), we have m(e^{π}r) ≥ π^{2}/3ℓ. Moreover, we get f_{γ}^{∗}h_{R}(1) ≥ h_{𝔸}(1) ≥ 1/4(1 − r) ≥ 1/4(1 − e^{−4π}), and then f_{γ}^{∗}h_{R}(1)^{2} ≥ 1/16(1 − e^{−4π})^{2} > 3/16π. So, now we can see (h_{R}/q_{R})(f_{γ}(1))^{2} ≥ π/8ℓ. Hence, we have 1/λ(R) ≤ max{8H(R)^{2}/π, 8/π}. Noting that h_{R}/q_{R} ≥ 1 always holds by Proposition 1.2, we conclude that 1/λ(R) ≤ CH(R)^{2} with C = 8/π = 2.5464 ….

Now we show the latter part. Suppose that, for a positive number λ, there are no constants K(λ) which satisfy the property in the statement above. Then there exist a sequence of Riemann surfaces R_{n} with λ(R_{n}) ≥ λ and a sequence of points p_{n} ∈ R_{n} (n = 1, 2, …) such that

\frac{h_{R_{n}}}{q_{R_{n}}} (p_{n}) → +∞

as n → ∞. Let G_{n} be a Fuchsian group acting on 𝔻 such that 𝔻/G_{n} = R_{n} and u_{n}(0) = p_{n}, where u_{n} : 𝔻 → 𝔻/G = R is the natural projection. Now we can use the following result due to Chabauty [2] (see also [5]).

Lemma 5.2. Let G_{n} be a sequence of Fuchsian groups acting on 𝔻. If there exists a neighborhood V of the identity element id in the Lie group Aut 𝔻 ≅ PSL(2, ℝ) such that V ∩G_{n} = {id} for all n. Then there exists a subsequence G_{n_{j}} which converges geometrically to a discrete subgroup G of Aut 𝔻.

Here we say that G_{n} geometrically converges to G if the following two conditions are satisfied:

1. For each γ ∈ G there exists a sequence of elements γ_{n} ∈ G_{n} such that γ_{n} → γ as n → ∞.

2. If γ_{n} ∈ G_{n} converges to an element γ of Aut 𝔻, then γ ∈ G.

We return to our situation. By this lemma, we may assume that G_{n} geometrically converges to some Fuchsian group G. Note also that G is torsion-free by assumption. Now let R = 𝔻/G and p = u(0), where u : 𝔻 → R is the natural projection. As is easily seen, λ(R) ≥ λ, in particular, R is non-exceptional. Therefore, by Lemma 1.1, we can find a φ ∈ A(R) which does not vanish at the point p. Then, by the surjectivity of the Poincaré series operator (see, for example, [6]), there exists a ψ ∈ A(𝔻) such that u^{∗}φ = Θ_{G}ψ. We may assume that ∥ψ∥_{A(𝔻)} = π. Set φ̃_{n} = Θ_{G_{n}}ψ and choose φ_{n} ∈ A(R_{n}) so that φ̃_{n} = u_{n}^{∗}φ_{n} for n = 1, 2, …. Then, by [11, Proposition A.2.4], φ̃_{n} tend to φ̃ uniformly on each compact set in 𝔻 as n → ∞. Since, φ̃(0) ≠ 0, we have |φ̃_{n}(0)| ≥ c_{0} for sufficiently large n, where c_{0} is a positive constant. Noting ∥φ_{n}∥_{A(R_{n})} ≤ ∥ψ∥_{A(𝔻)} = π, we obtain u_{n}^{∗}q_{R_{n}}(0)^{2} ≥ |φ̃_{n}(0)| ≥ c_{0} for sufficiently large n. On the other hand, by Lemma 1.5, u_{n}^{∗}h_{R_{n}}(0) ≥ u_{n}^{∗}ρ_{R_{n}}(0) coth(λ)/4 = coth(λ)/4. Thus we have arrived at the contradiction (h_{R_{n}}/q_{R_{n}})(p_{n})^{2} ≤ coth^{2}(λ)/16c_{0} for n large enough.

Remark. As Proposition 4.7 suggests, we may state the conjecture: H(R)^{2} ≤ C max{λ(R)^{−1}, 1} for some absolute constant C. The only one obstacle to employ the same method as above is the presence of the case that R_{n} tends to the exceptional Riemann surface, say, ℂ − {0, 1}. This might be overcome by a refinement of estimate of q_{R}, when Proposition 4.7 would be of use.

6. A lower estimate of q_{R} in terms of the logarithmic capacity

It seems quite hard to control the metric q_{R} from below in general. In this section, we propose a lower estimate of q_{R} by using the Poincaré metric ρ_{R} and the capacity metric c_{R} in the case that R ∉ O_{G}, i.e., R carries the Green function. Throughout this section, we always assume that R ∉ O_{G}.

We denote by c_{R} the capacity metric (or the logarithmic capacity) of R; precisely, letting α be a local coordinate of R, the metric c_{R} is defined by the relation

G_{R}(p, p_{0}) = − log |α(p) − α(p_{0})| − log c_{R,α}(z_{0}) + o(1)

as p → p_{0}, where z_{0} = α(p_{0}) and G_{R}(p, p_{0}) is the Green function of R with pole at p_{0}. Classically, − log c_{R} is called the Robin constant.

Our result is as follows.

Theorem 6.1. For R ∉ O_{G}, the inequality q_{R}^{2} ≥ ρ_{R}c_{R} holds.

Proof. Fix p_{0} ∈ R and take a holomorphic universal covering u : 𝔻 → R from the unit disk 𝔻 with covering transformation group G so that u(0) = p_{0}. Note that G is a Fuchsian group of convergence type. In particular, the Blaschke product

B(z) = ∏_{g∈G−\{1\}} \frac{|g(0)|}{−g(0)} \frac{z − g(0)}{1 − \overline{g(0)}z}

converges in 𝔻. Also note that u^{∗}c_{R}(0) = B(0) (see [17]). Set φ = Θ_{G}B. Then we have φ(0) = B(0) = u^{∗}c_{R}(0) = u^{∗}ρ_{R}(0)u^{∗}c_{R}(0). Since ∥φ∥_{A(𝔻,G)} ≤ ∥B∥_{A(𝔻)} ≤ π, we obtain u^{∗}q_{R}(0)^{2} ≥ |φ(0)| = u^{∗}ρ_{R}(0)u^{∗}c_{R}(0), which means q_{R}^{2} ≥ ρ_{R}c_{R} at p_{0}.

7. Relationship with the kernel function for A(R)

We explain the magnitude of q_{R} can be described approximately by the usual kernel function for A(R). First, we recall necessary definitions. Let (φ, ψ)_{R} be the natural pairing between A(R) and B(R), namely,

(φ, ψ)_{R} = \frac{3}{π} ∫∫ _{R} ρ_{R}^{−2}φ\overline{ψ} = \frac{3}{π} ∫∫ _{R} ρ_{R}(ζ)^{−2}φ(ζ)\overline{ψ(ζ)}dξdη

for φ ∈ A(R) and ψ ∈ B(R). (The extra factor 3/π is only for convention.)

Set K_{𝔻}(ζ, z) = (1 − z̄ζ)^{−4} for ζ, z ∈ 𝔻. Let G be a torsion-free Fuchsian group acting on the unit disk 𝔻 with 𝔻/G = R. Then we set

F (ζ, z) = ∑_{g∈G} K_{𝔻}(g(ζ), z)g′(ζ)^{2}

for z and ζ ∈ 𝔻. We denote by u the canonical projection 𝔻 → 𝔻/G = R and define the kernel function K_{R}(r, p) as a section of the line bundle K_{R}^{⊗2} ⊠ \overline{K}_{R}^{⊗2} over R × R which is determined by

(u × u)^{∗}K_{R}(ζ, z) = F (ζ, z)

for ζ, z ∈ 𝔻. In other words, K_{R}(u(ζ), u(z))u′(ζ)^{2}\overline{u′(z)^{2}} = F (ζ, z) holds locally.

Proposition 7.1 ([6] or [3]). The kernel function K_{R}(r, p) is holomorphic in r and antiholomorphic in p and satisfies the following properties.

1. K_{R}(r, p) = \overline{K_{R}(p, r)},

2. ∥K_{R}(·, p)∥_{A(R)} = ∫∫_{R} |K_{R}(ζ, p)|dξdη ≤ πρ_{R}(p)^{2}, where ζ = ξ + iη,

3. K_{R}(·, p) = K_{R}(ζ, p)dζ^{2} ∈ B(R) for each fixed p ∈ R, and

4. φ(p) = (φ, K_{R}(·, p))_{R} for each φ ∈ A(R).

We set μ_{R}(p) = ∥K_{R}(·, p)∥_{B(R)} = sup_{r∈R} ρ_{R}(r)^{−2}|K_{R}(r, p)|. Then, from the above properties, it can be deduced that μ_{R} = μ_{R}(z)|dz|^{2} is a (1, 1)-form on R and the next result follows.

Theorem 7.2. The inequality μ_{R} ≤ q_{R}^{2} ≤ 3μ_{R} is valid on R for an arbitrary nonexceptional hyperbolic Riemann surface R.

Proof. By property 2 above, we immediately have μ_{R}(p) ≤ q_{R}^{2}(p). On the other hand, by property 4, we have |φ(p)| ≤ \frac{3}{π}μ_{R}(p)∥φ∥_{A(R)}. This implies q_{R}(p)^{2} ≤ 3μ_{R}(p).

Setting M(R) = sup_{p∈R}(μ_{R}/ρ_{R}^{2})(p), we have the following

Corollary 7.3.

M(R) ≤ πκ(R) ≤ 3M(R).

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

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