Metrics of equicontinuity for riemann surfaces

Abstract

We write δ_t(ε) for the modulus of continuity at t∈Δ of any holomorphic cover of a Riemann surface Z by the unit disk Δ That modulus is computed in terms of the absolute value metric for Δ and any metric σ a for Z's topology. THEOREM As t goes to the rim of Δ, the rate at which δ_t(ε) goes to zero satisfies, for each ε the restriction

if and only if every family Hol(X, Z) is σ-equicontinuous {i.e. σ is a metric of equicontinuity for Z). In particular, that rate's dependence on σ satisfies the restriction

when σ = dz, the Kobayashi distance (known to be a metric of equicontinuity when Z is hyperbolic). With Z ≡ Δ and σ as the absolute value metric, we also compute the minimum of the moduli at t of all biholomorphisms B of Δ to be

for 0<ε<1 That minimum coincides with the moduli at t of those B with |B(t)|= ε/2.

AMS(MOS)::

• 30F99
• 32H20