Hyperbolic Metric and Hypergeometric Functions

Toshiyuki Sugawa

Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526 Japan

          sugawa@math.sci.hiroshima-u.ac.jp

January 7, 2003, Kurashiki

Hyperbolic Metric. The hyperbolic metric of a (hyperbolic) Riemann surface R is a complete conformal metric ρ_{R}(z)|dz| on R satisfying the equation

Δ log ρ_{R} = 4ρ_{R}^{2}.

If we write ρ_{R}(z) = e^{φ(z)/2}, the equation is equivalent to the Liouville equation

Δφ = 8e^{φ}.

If z = 0 is a puncture of R, the density φ(z) behaves as

φ(z) = 2 log \frac{1}{|z|} − 2 log log \frac{1}{|z|} + O(1).

Example. If we write 𝔻 = {z ∈ ℂ; |z| < 1}, we have

ρ_{𝔻}(z) = \frac{1}{1 − |z|^{2}} .

Branch Singularities. If w = z^{n} for an integer n ≥ 2, the equation

ρ(w) |\frac{dw}{dz}| = ρ_{𝔻}(z)

gives the solution

ρ(w) = \frac{1}{n|w|^{1−1/n}(1 − |w|^{2/n})}

= \frac{1}{n}|w|^{\frac{1}{n} −1}(1 + o(1)) (w → 0).

Therefore, φ = 2 log ρ has the asymptotic behaviour like

φ(w) = 2 (1 − \frac{1}{n}) log \frac{1}{|w|} + O(1)

as w → 0. Geometrically, the metric ρ(w) is obtained by indentifying the both sides of the hyperbolic sector with opening 2π/n. The same idea may apply to the case when n is not necessarily an integer.

Conical Singularities. A conformal metric ρ(z)|dz| on a Riemann surfaces R is said to have conical singularity at p ∈ R with order α > 0 if φ = 2 log ρ satisfies

φ(z) = 2(1 − α^{−1}) log \frac{1}{|z|} + O(1)

as z → 0 where z is a local chart around p with z(p) = 0. For this notion, see, for example, [5] and the references therein.

A function α_{R} : R → [0, +∞) is called a singularity data if the set of “singularities”

Sing(α_{R}) = {p ∈ R; α_{R}(p) ≠ 1}

is a discrete subset of R.

In this article, we will allow a conformal metric ρ = ρ(z)|dz| on R to have a (closed) discrete subset where the density may be +∞ or 0. A conformal metric ρ on R (in the above sense) is called the hyperbolic metric of R with singularity data α_{R} if the induced distance is complete, if φ = 2 log ρ satisfies the Liouville equation on R \ Sing(α_{R}), and if

φ(z) = 2(1 − α_{R}(p)^{−1}) log \frac{1}{|z|} + O(1)

for each p ∈ Sing(α_{R}), where z is a local chart around p with z(p) = 0.

The Gauss-Bonnet formula gives the next hyperbolicity condition for α_{R} if there is a hyperbolic metric on R with it as singularity data:

∑_{p∈Sing(α_{R})}(1 − α_{R}(p)^{−1}) > 2.

When the above inequality is satisfied, α_{R} is called hyperbolic.

E. Picard [3] proved the existence of the hyperbolic metric with conical singularities when R is the Riemann sphere. Much later, Heins [1] proved it for general Riemann surfaces.

Altough Poincaré’s approach using the Fuchsian differential equations could give good information on the metric, it seems that no concrete forms of the hyperbolic metric with conical singularities are known.

Hypergeometric Functions. The Gauss’ hypergeometric function is defined by

F (a, b, c; z) = ∑_{n=0}^{∞} \frac{(a)_{n}(b)_{n}}{(c)_{n}n!} z^{n},

where (a)_{n} = a(a + 1) ⋯ (a + n − 1) denotes the shifted factorial, for c ≠ −1, −2, ….

The Case of R = ℂ̂ and Sing(α_{R}) = {0, 1, ∞}. Assume that αℂ̂(0) = α, αℂ̂(1) = β, αℂ̂(∞) = γ and that the hyperbolicity condition

\frac{1}{α} + \frac{1}{β} + \frac{1}{γ} < 1

holds. Set

a = (1 − α − β − γ)/2,
b = (1 − α + β − γ)/2,
c = 1 − α.

Then, the Schwarz triangle function

H(z) = \frac{z^{1−c}F (a − c + 1, b − c + 1, 2 − c; z)}{F (a, b, c; z)}

maps the upper half-plane onto the domain bounded by two segments and a circular arc forming the angles πα, πβ, πγ at the vertices (see [2, p. 315]). Note also that

H(1) = \frac{Γ(2 − c)Γ(c − a)Γ(c − b)}{Γ(c)Γ(1 − a)Γ(1 − b)}.

Choose an r > 0 so that the circular arc intersects perpendicularly the circle |z| = r. Then H(z)/r maps the upper half-plane onto the hyperbolic triangle in 𝔻 with angles πα, πβ, πγ. Thus, the hyperbolic metric with the given singularity data is given by the pull-back of ρ_{𝔻} = |dz|/(1 − |z|^{2}) by H(z)/r, namely,

ρ(z) = \frac{r|H′(z)|}{r^{2} − |H(z)|^{2}} .

Special Case α = β = γ. In this case,

r = \frac{H(1)}{\sqrt{cos πα}}

= \frac{1}{\sqrt{π cos πα}} · \frac{Γ(1 + α)Γ((1 − α)/2)}{Γ(1 − α)Γ((1 + α)/2)Γ(1/2 + α)}.

For the proof of the above formula, we use the second cosine theorem

cosh C = \frac{cos πα cos πβ + cos πγ}{sin πα sin πβ} ,

where C is the hyperbolic length of the circular arc in 𝔻 (see, for instance, [4, Theorem 8.5]).

For a general case, one can compute the radius r possibly with much efforts.

References

[1] M. Heins, On a class of conformal metrics, Nagoya J. Math. 21 (1962), 1–60.

[2] Z. Nehari, Conformal Mappings, McGraw-Hill, New York, 1952.

[3] E. Picard, De l’intégration de l’équation Δu = e^{u} sur une surface de Riemann fermeé, Crelle’s J. 130 (1905), 243–258.

[4] S. Stahl, The Poincaré Half-Plane, Jones and Bartlett, Boston and London, 1993.

[5] L. Takhtajan and P. Zograf, Hyperbolic 2-spheres with conical singularities, accessory parameters and Kähler metrics on M_{0,n}, to appear in Tran. Amer. Math. Soc.