VARIOUS GEOMETRIC ASPECTS OF

UNIFORMLY PERFECT DOMAINS

一様完全領域の幾何学的諸相

Toshiyuki Sugawa 須川敏幸

1995/11/17 山口大学

§1. Uniform perfectness of the hyperbolic Riemann surface

In this talk, we shall consider only the hyperbolic Riemann surfaces R endowed with the Poincaré metric ρ_{R}(z)|dz| of constant negative curvature −4. We denote by D_{R}(p, r) the hyperbolic disk in R centered at p ∈ R of radius r. We set σ(p) = σ_{R}(p) = sup{r > 0; D_{R}(p, r) is simply connected } and H_{R} = inf_{p∈R} σ_{R}(p), which is called the injectivity radius of R.

According to [LM], R is called uniformly perfect if its injectivity radius H_{R} is positive (including infinity).

Let R_{R} be the set of essential ring domains in R, where ring domain R_{0} is essential if the inclusion map R_{0} ↪ R is π_{1}-injective. The module m(R_{0}) of R_{0} ∈ R_{R} is defined by the number m such that R_{0} is conformally equivalent to the annulus {z ∈ ℂ; 1 < |z| < e^{m}}. The core curve of R_{0} ∈ R_{R}, denoted by Core(R_{0}), is the unique simple closed geodesic of R_{0} (with complete hyperbolic metric).

On R, another important continuous metric ρ̂_{R}, called the Hahn metric, is defined by

ρ̂_{R}(z)|dz| = inf_{G} ρ_{G}(z)|dz|,

where G ranges over all simply connected domains with p ∈ G and z is a fixed local coodinate around p ∈ R. By the monotoneity of the Poincaré metric, ρ̂_{R} ≥ ρ_{R}.

We set M_{R} = sup_{R_{0}∈R_{R}} m(R_{0}) and K_{R} = sup_{p∈R} \frac{ρ̂_{R}}{ρ_{R}} (p). Now we have the following estimates. (The part (2) is due to Gotoh [G].)

Theorem 1.1.

(1) 2H_{R} ≤ π^{2}/M_{R} ≤ 2H_{R}e^{2H_{R}}.

(2) \frac{1}{4} coth H_{R} ≤ K_{R} ≤ coth H_{R}.

Corollary 1.2.

The following conditions are mutually equivalent.

(1) R is uniformly perfect (i.e., H_{R} > 0),

(2) M_{R} < +∞,

Typeset by AMS-TEX

(3) K_{R} < +∞.

We shall close this section by exhibiting a simple application of uniform perfectness. Let A_{2}(R) and B_{2}(R) be the complex Banach spaces of holomorphic quadratic differentials φ on R with norms ∥φ∥_{1} = ∫∫_{R} |φ(z)|dxdy and ∥φ∥_{∞} = sup_{R} |φ|ρ_{R}^{−2}, respectively. We set κ_{R} = sup{∥φ∥_{∞}; φ ∈ A_{2}(R), ∥φ∥_{1} = 1}.

Theorem 1.3.

κ_{R} ≤ \frac{1}{π} coth^{2} H_{R}.

In particular, A_{2}(R) ⊂ B_{2}(R), if R is uniformly perfect.

Remark. Matsuzaki [M] proved this theorem in a sharper form, and with full generality. By our result, we see that κ_{R} = O(H_{R}^{−2}) as H_{R} → 0, but, in fact, κ_{R} = O(H_{R}^{−1}) as H_{R} → 0 by an argument using the Marden-Marglis constant (see [M]).

Proof of Theorem 1.3. Fix an arbitrary point p in R. Let π : Δ = {|z| < 1} → R be a holomorphic universal covering map with π(0) = p. We denote by φ̃ the pullback of φ ∈ A_{2}(R) by π. Then |φρ_{R}^{−2}|(p) = |φ̃(0)| by the conformal invariance of the differential forms. On the other hand, for r = tanh(σ(p)), by the mean value property, we have

φ̃(0) = \frac{1}{πr^{2}} ∫∫_{|z|<r} φ̃(z)dxdy

Since π is injective in D_{Δ}(0, σ(p)), we have

|φρ_{R}^{−2}|(p) = |φ̃(0)| ≤ \frac{1}{πr^{2}} ∫∫_{|z|<r} |φ̃(z)|dxdy

≤ \frac{1}{πr^{2}} ∫∫_{R} |φ| = \frac{1}{πr^{2}} ∥φ∥_{1} ≤ \frac{1}{π} coth^{2} H_{R} · ∥φ∥_{1}.

Thus we have the assertion that ∥φ∥_{∞} ≤ \frac{1}{π} coth^{2} H_{R} · ∥φ∥_{1}. □

§2. Hyperbolic and extremal lengths

We denote by S_{R} the set of all free homotopy classes of non-trivial simple closed loops in R. The hyperbolic length ℓ[α] of [α] ∈ S_{R} is defined by

ℓ[α] = inf_{α′∈[α]} ∫_{α′} ρ_{R}(z)|dz|.

Let π : Δ → R be a holomorphic universal covering map of R and Γ its covering transformation group. If an element γ ∈ Γ covers [α] ∈ S_{R}, then we have |trγ| = 2 cosh ℓ[α] (where |trγ| denotes the absolute value of the trace of an element of SL(2, ℝ) representing γ).

Thus we can easily see that

H_{R} = \frac{1}{2} inf_{[α]∈S_{R}} ℓ[α]

and

2 cosh(2H_{R}) = inf_{γ∈Γ \{1}} |trγ|.

The extremal length E[α] of [α] ∈ S_{R} is defined by

E[α] = sup_{τ} \frac{(inf_{α′∈[α]} ∫_{α′} τ(z)|dz|)^{2}}{∫∫_{D} τ(z)^{2}dxdy} ,

where the supremum is taken over all Borel measurable conformal metrics τ = τ(z)|dz| on R. As for this, the following result due to Jenkins-Strebel is fundamental.

Theorem 2.1 (cf.[St]). For any [α] ∈ S_{R} with E[α] > 0, there exists an integrable holomorphic quadratic differential φ_{0} (Jenkins-Strebel differential) with closed trajectries homotopic to α and whose characteristic ring domain R_{0} ∈ R_{R} satisfies the following conditions.

(1) E[α] = \frac{(inf_{α′∈[α]} ∫_{α′} |φ|^{1/2}|dz|)^{2}}{∫∫_{R} |φ|dxdy} ,

(2) m(R_{0}) = \frac{2π}{E[α]},

(3) m(R_{1}) ≤ m(R_{0}) for all R_{1} ∈ R_{R} with Core(R_{1}) ∈ [α].

Corollary 2.2.

inf_{[α]∈S_{R}} E[α] = \frac{2π}{M_{R}} .

The following theorem connects amounts of the hyperbolic and extremal lengths, and from which we can directly deduce Theorem 1.1 (1).

Theorem 2.3.

\frac{2}{π} ℓ[α] ≤ E[α] ≤ \frac{ℓ[α]}{arctan(\frac{1}{sinh ℓ[α]})}.

By an elementary calculation, we know that \frac{π}{2} < e^{x} arctan(\frac{1}{sinh x}) < 2 for any x > 0, we have the next

Corollary 2.4.

\frac{2}{π} ℓ[α] ≤ E[α] ≤ \frac{2}{π} ℓ[α]e^{ℓ[α]}.

Maskit showed the similar result that \frac{2}{π} ℓ[α] ≤ E[α] ≤ ℓ[α]e^{ℓ[α]} in [Mas].

On the other hand, Matsuzaki [M] showed the next

Theorem 2.5.

E[α] ≤ κ_{R}ℓ[α]^{2}.

Proof. Let φ_{0} be the holomorphic diffrential on R with ∥φ_{0}∥_{1} = 1 which gives an extremal metric |φ_{0}|^{1/2}|dz| as in Theorem 2.1. Then, for α′ ≃ α,

E[α]^{1/2} ≤ ∫_{α′} |φ_{0}|^{1/2}|dz| = ∫_{α′} |φ_{0}ρ_{R}^{−2}|^{1/2} · ρ_{R}|dz| ≤ ∥φ_{0}∥_{∞}^{1/2} ∫_{α′} ρ_{R}|dz|.

Since α′ is arbitrary, we obtain that E[α] ≤ ∥φ_{0}∥_{∞}ℓ[α]^{2}. □

By combining Theorem 1.3 with the theorem above, we have the next Corollary 2.6. E[α] ≤ \frac{1}{π} coth^{2} H_{R} · ℓ[α]^{2}.

Remark. Of course, by a refined result of Matsuzaki [M], we shall have a better estimate than the above.

Finally, we refer to the quasi-invariance of these amounts. Let f : R → R′ be a K-quasiconformal homeomorphism, and set α′ = f(α). Then, it is clear that E[α]/K ≤ E[α′] ≤ KE[α]. Moreover it also holds that ℓ[α]/K ≤ ℓ[α′] ≤ Kℓ[α] (see Wolpert [W]). In particular, we know that H_{R}/K ≤ H_{R′} ≤ KH_{R} and M_{R}/K ≤ M_{R′} ≤ KM_{R}.

§3. Uniformly perfect plane domains

As we have seen in the previous sections, the uniform perfectness can be defined by the intrinsic hyperbolic geometry of the surface. But, the uniform perfectness seems to have its most importance in plane domains. The various equivalent definitions of uniform perfectness for plane domains tell us the richness of this notion.

In the sequel, let D be a subdomain of ℂ̂ with #(ℂ̂ \ D) ≥ 3. And, let π : Δ → D be a holomorophic universal covering map. We set N_{D} = ∥S_{π}∥_{Δ} := sup_{z∈Δ} |S_{π}(z)|(1 − |z|^{2})^{2}, where S_{π} = (π′′/π′)′ − \frac{1}{2}(π′′/π′)^{2} is the Schwarzian derivative of π. Note that N_{D} does not depend on the particular choice of π.

By the Nehari-Kraus theorem, we know that N_{D} ≤ 6 if D is simply connected. Now we state the supplementary result concerning with N_{D}.

Theorem 3.1 (Minda [Mi]). If D is not simply connected, we have

\frac{π^{2}}{2H_{D}} + 2 ≤ N_{D} ≤ 6 coth^{2} H_{D}.

Let A_{D} denote a subclass of R_{R} consisting of all round annuli, and set A_{D} := sup_{R_{0}∈A_{D}} m(R_{0})(≤ M_{D}). Then, we can show the following result.

Theorem 3.2 (cf. McMullen [Mc]). If D ⊂ ℂ, it holds that M_{D} ≤ A_{D} + 5 log 2.

In case of ∞ ∈ D, we have the next auxiliary result.

Theorem 3.3. If L ∈ Möb, \frac{1}{2}A_{L(D)} − log 4/3 ≤ A_{D}.

If D ⊂ ℂ, further we define the domain constant

c_{D} = inf_{z∈D} δ_{D}(z)ρ_{D}(z),

where δ_{D}(z) = dist(z, ∂D) = inf_{a∈∂D} |z − a|.

That is, c_{D} is the infimum of the ratio of the Poincaré metric ρ_{D}(z)|dz| to the quasi-hyperbolic one |dz|/δ_{D}(z). We should note that δ_{D}(z)ρ_{D}(z) ≤ 1 for any z ∈ D, thus c_{D} ≤ 1. Concerning this, the similar result as Theorem 1.1 (2) is verified.

Theorem 3.4 (Minda [Mi]).

\frac{tanh H_{D}}{4} ≤ c_{D} ≤ \frac{2\sqrt{3}}{π}H_{D}.

Remark. The assumption that ∞ ∉ D is essential for c_{D}. In fact, if D = Δ^{∗} = ℂ̂ \ \overline{Δ}, we have δ_{Δ^{∗}}(z) = |z| − 1 and ρ_{Δ^{∗}}(z) = \frac{1}{|z|^{2}−1}, therefore δ_{Δ^{∗}}(z)ρ_{Δ^{∗}}(z) = \frac{1}{|z|+1} → 0 as z → ∞, whereas the Möbius equivalent domain Δ satisfies that c_{Δ} = \frac{1}{2} > 0.

Finally, we shall summalize our results.

Theorem 3.4. Let D be a plane domain of hyperbolic type. Then the following conditions are mutually equivalent.

(1) H_{D} > 0,

(2) M_{D} < ∞,

(3) A_{D} < ∞,

(4) N_{D} < ∞,

(5) c_{D} > 0 (if D ⊂ ℂ).

The other features of uniformly perfect domains can be seen in [Pom1] and [Pom2].

References

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Kyoto Univ. 35 (1995), 299–324. [LM]. X. Liu, D. Minda, Monotonicity of hyperbolic curvature under univalent mappings, Ann.

Acad. Sci. Fenn. Ser. A. I. Math. 16 (1991), 227–242. [Mas]. B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A. I.

Math. 10 (1985), 381–386. [M].K. Matsuzaki, Geometric Complex Analysis, edited by J. Noguchi et al, World Scientific,

Singapore, 1995. [Mc].C. McMullen, Complex Dynamics and Renormalization, Ann. of Math. Studies, Princeton,

1994. [Mi].D. Minda, The Schwarzian derivative and univalence criteria, Contemporary Math. 38 (1985),

45–52. [Pom1]. Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. 32 (1979),

192-199. [Pom2]. Ch. Pommerenke, On uniformly perfect sets and Fuchsian groups, Analysis 4 (1984), 299-321. [St].K. Strebel, Quadratic Differentials, Springer, 1984. [W].S. Wolpert, The length spectra as moduli for compact Riemann surfaces, Ann. of Math. 109

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