Lipschitz constants for the real part and modulus of analytic mappings on a negatively curved surface
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Archiv der Mathematik
Lipschitz constants for the real part and modulus of analytic mappings on a negatively curved surface ´ Marijan Markovic Abstract. We prove that if f , |f | < 1, is an analytic mapping on a surface Σ with curvature bounded from below by a constant k < 0, and if σ is the hyperbolic distance on the unit disc, we have 4k dΣ (ζ, η), ζ ∈ Σ, η ∈ Σ, π where dΣ is the distance on Σ generated by a conformal metric on Σ. On the other hand, if |f | < 1, then σ(f (ζ), f (η)) ≤ −
σ(|f |(ζ), |f |(η)) ≤ −kdΣ (ζ, η),
ζ ∈ Σ, η ∈ Σ.
Mathematics Subject Classification. Primary 31C05, 32Q05; Secondary 30A10. Keywords. Analytic mappings, Harmonic functions, Negatively curved surfaces, Hyperbolic distance.
1. Introduction. It is well known that all conformal automorphisms of the unit disc U = {z ∈ C : |z| < 1} onto itself are of the form eiα ϕz , where α ∈ R and z−w , 1 − zw The hyperbolic distance on U is given by ϕz (w) =
σ(z, w) = 2 log
w ∈ U.
|1 − zw| + |z − w| 1 + |ϕz (w)| = 2 log , 1 − |ϕz (w)| |1 − zw| − |z − w|
z ∈ U, w ∈ U.
The classical Schwarz–Pick inequality states that for an analytic function f of the unit disc into itself, we have σ(f (ζ), f (η)) ≤ σ(ζ, η),
ζ ∈ U, η ∈ U.
´ M. Markovic
Arch. Math.
This is equivalent to |f (ζ)| ≤
1 − |f (ζ)|2 , 1 − |ζ|2
ζ ∈ U.
In other words, an analytic mapping of the unit disc into itself does not increase the hyperbolic distance. This classical result has numerous generalizations. We refer to the result by Ahlfors [1,2] for surfaces, and the very general result by Yau for manifolds [5,9]. The result by Yau (in a special case) for surfaces is given in Proposition 2.2 below. Recently Kalaj and Vuorinen [4,8] proved that a real valued harmonic function of the unit disc into itself does not increase the hyperbolic distance by a factor greater than 4/π. More precisely, if U is a harmonic function of the unit disc U into the interval (−1, 1), then 4 σ(ζ, η), ζ ∈ U, η ∈ U. π On the other hand, Pavlovi´c [7,8] showed a result concerning the modulus of an analytic mapping F : U → U which states that σ(U (ζ), U (η)) ≤
ρ(|F |(ζ), |F |(η)) ≤ ρ(ζ, η),
ζ ∈ U, η ∈ U,
where ρ(z, w) = |ϕz (w)|,
z ∈ U, w ∈ U,
is the pseudohyperbolic distance. The main aim of this paper is to consider these results for analytic mappings on surfaces with negative curvature with an image in the unit disc. 2. Preliminaries. Let Σ ⊆ Rm be a domain. A conformal metric on Σ is a positive C 2 -continuous function on Σ. The distance on Σ generated by a conformal metric λΣ is dΣ (ζ, η) = inf λΣ (ζ)|dζ|, γ
γ
where γ is among partially smooth curves on Σ that connect the points ζ ∈ Σ and η ∈ Σ. A surface is a domain in C equipped with a conformal metric. The curvature of a surface Σ with a conformal metric λΣ is kΣ (ζ) = −
Δ log λΣ (ζ) , λ2Σ (ζ)
ζ ∈ Σ.
We consider the unit disc U with the hyperbolic metric 2 , z ∈ U, λU (z) = 1 − |z|2 which is normalized such that it has constant curvature equal to −1 [2]. The corresponding distance between z ∈ U and w ∈ U is the hyperbolic dis
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