The Hyperbolic Metric In Complex Analysis

Geometry of conformal metricsThe hyperbolic metric on an arbitrary domainHyperbolic metricDefinitionLet D be a domain in the plane. The hyperbolic metric of D is theunique complete metric on D with constant negative curvature 4(provided that it exists).Example: The hyperbolic metric on D is λ(z) 1/(1 z 2 ).Example: The hyperbolic metric on DR isλR (z) R2R. z 2Why? Because λR is the pull-back of the hyperbolic metric, and so it iscomplete and constant curvature 4.Eric SchippersHyperbolic metricOctober 15, 201523 / 41

Geometry of conformal metricsThe uniformization theoremUniformization theoremTheorem (Uniformization theorem (Koebe, Poincaré))Every simply connected Riemann surface is biholomorphicallyequivalent to the Riemann sphere C, the complex plane C, or the unitdisk D.Eric SchippersHyperbolic metricOctober 15, 201524 / 41

Geometry of conformal metricsThe uniformization theoremUniformization theoremTheorem (Uniformization theorem (Koebe, Poincaré))Every simply connected Riemann surface is biholomorphicallyequivalent to the Riemann sphere C, the complex plane C, or the unitdisk D.Proof.Too much work for this talk.Eric SchippersHyperbolic metricOctober 15, 201524 / 41

Geometry of conformal metricsThe uniformization theoremUniformization theoremTheorem (Uniformization theorem (Koebe, Poincaré))Every simply connected Riemann surface is biholomorphicallyequivalent to the Riemann sphere C, the complex plane C, or the unitdisk D.Proof.Too much work for this talk.CorollaryEvery Riemann surface is given by the quotient of C, C or D by a nicegroup action (think tiling).Eric SchippersHyperbolic metricOctober 15, 201524 / 41

Geometry of conformal metricsThe uniformization theoremAlmost everything is covered by D C only covers C C only covers C, C\{0} (cylinder) and tori.Eric SchippersHyperbolic metricOctober 15, 201525 / 41

Geometry of conformal metricsThe uniformization theoremAlmost everything is covered by D C only covers C C only covers C, C\{0} (cylinder) and tori. Everything else is D/G for some nice subgroup of the Möbiustransformations of the formT (z) eiθz a1 āza D.which are hyperbolic isometries. So almost every Riemann surfacehas a hyperbolic metric inherited from .Eric SchippersHyperbolic metricOctober 15, 201525 / 41

Geometry of conformal metricsThe uniformization theoremUniformization theoremNearly all domains have a hyperbolic metric.Corollary (Uniformization theorem)Any subset of the plane which omits at least two points possesses ahyperbolic metric.Eric SchippersHyperbolic metricOctober 15, 201526 / 41

Complex analysisIntroductionWhere’s the complex analysis?The isometries of the hyperbolic metric are exactly the conformalautomorphisms of the disc.(Conformal automorphisms one-to-one, onto analytic maps)Eric SchippersHyperbolic metricOctober 15, 201527 / 41

Complex analysisIntroductionWhere’s the complex analysis?The isometries of the hyperbolic metric are exactly the conformalautomorphisms of the disc.(Conformal automorphisms one-to-one, onto analytic maps)So there should be some connection between complex analysis on thedisc and hyperbolic geometry on the disc.Let’s look at some examples.Eric SchippersHyperbolic metricOctober 15, 201527 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaSchwarz lemmaTheoremIf f : D D is analytic and f (0) 0 then f (z) z .Eric SchippersHyperbolic metricOctober 15, 201528 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaSchwarz lemmaTheoremIf f : D D is analytic and f (0) 0 then f (z) z .The “hyperbolically correct” version isTheorem (hyperbolic Schwarz lemma)If f : D D is analytic thend(f (z), f (w)) d(z, w).Eric SchippersHyperbolic metricOctober 15, 201528 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaSchwarz lemmaTheoremIf f : D D is analytic and f (0) 0 then f (z) z .The “hyperbolically correct” version isTheorem (hyperbolic Schwarz lemma)If f : D D is analytic thend(f (z), f (w)) d(z, w).Analytic maps from D to D are contractions.Eric SchippersHyperbolic metricOctober 15, 201528 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaproof of hyperbolic Schwarz lemmaLetT (z) z w,1 w̄zS(ζ) ζ f (w)1 f (w)ζ.So if f : D D then S f T (0) S(f (w)) 0.Eric SchippersHyperbolic metricOctober 15, 201529 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaproof of hyperbolic Schwarz lemmaLetT (z) z w,1 w̄zS(ζ) ζ f (w)1 f (w)ζ.So if f : D D then S f T (0) S(f (w)) 0.By the Schwarz lemma, S(f (T (z))) z S(f (z)) T 1 (z) sof (z) f (w)1 f (w)f (z)Eric Schippers z w.1 w̄zHyperbolic metricOctober 15, 201529 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaproof of hyperbolic Schwarz lemmaLetT (z) z w,1 w̄zS(ζ) ζ f (w)1 f (w)ζ.So if f : D D then S f T (0) S(f (w)) 0.By the Schwarz lemma, S(f (T (z))) z S(f (z)) T 1 (z) sof (z) f (w)1 f (w)f (z) z w.1 w̄zBut arctanh is increasing sod(f (z), f (w)) arctanhEric Schippersf (z) f (w)1 f (w)f (z) arctanhHyperbolic metricz w d(z, w).1 w̄zOctober 15, 201529 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaAhlfors’ generalization of the Schwarz lemmaTheorem (Ahlfors-Schwarz lemma, special case)Let DR be the disc of radius R, with hyperbolic metric λR . For anymetric ρ on DR , such that the curvature Kρ (z) 4 for all z,ρ(z) λR (z)for all z.Eric SchippersHyperbolic metricOctober 15, 201530 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaThe proofProof.For r R we have Dr DR . Letv (z) ρ;λrz Dr .v is continuous, positive, and v 0 as z r .Eric SchippersHyperbolic metricOctober 15, 201531 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaThe proofProof.For r R we have Dr DR . Letv (z) ρ;λrz Dr .v is continuous, positive, and v 0 as z r .So v has a maximum; thus log v has a maximum; say at z0 Dr .Eric SchippersHyperbolic metricOctober 15, 201531 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaThe proofProof.For r R we have Dr DR . Letv (z) ρ;λrz Dr .v is continuous, positive, and v 0 as z r .So v has a maximum; thus log v has a maximum; say at z0 Dr .0 4 log v (z0 ) 4 log ρ 4 log λr ρ2 (z0 )Kρ (z0 ) λ2r (z0 )Kλr (z0 ) 4ρ2 (z0 ) 4λ2r (z0 ).Eric SchippersHyperbolic metricOctober 15, 201531 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaThe proofProof.For r R we have Dr DR . Letv (z) ρ;λrz Dr .v is continuous, positive, and v 0 as z r .So v has a maximum; thus log v has a maximum; say at z0 Dr .0 4 log v (z0 ) 4 log ρ 4 log λr ρ2 (z0 )Kρ (z0 ) λ2r (z0 )Kλr (z0 ) 4ρ2 (z0 ) 4λ2r (z0 ).So since z0 was the maximum, ρ(z) λr (z) for all z.Eric SchippersHyperbolic metricOctober 15, 201531 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaThe proofProof.For r R we have Dr DR . Letv (z) ρ;λrz Dr .v is continuous, positive, and v 0 as z r .So v has a maximum; thus log v has a maximum; say at z0 Dr .0 4 log v (z0 ) 4 log ρ 4 log λr ρ2 (z0 )Kρ (z0 ) λ2r (z0 )Kλr (z0 ) 4ρ2 (z0 ) 4λ2r (z0 ).So since z0 was the maximum, ρ(z) λr (z) for all z.Now let r R.Eric SchippersHyperbolic metricOctober 15, 201531 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaAhlforsAhlfors 1907-1996According to Ahlfors: “This is an almost trivial fact and anyone whosees the need could prove it at once”.Eric SchippersHyperbolic metricOctober 15, 201532 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaAhlfors continuedFinnish mathematician, advisors at University of Helsinki were E.Lindelöf and R. NevanlinnaFirst Fields Medal (with Jesse Douglas) in 1936 for work in valuedistribution theory (Nevanlinna theory).Wolf Prize in 1981Towering figure in Riemann surfaces and complex analysisMost famous as one of the founders of Teichmüller theory andquasiconformal mappingsEric SchippersHyperbolic metricOctober 15, 201533 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaAhlfors’ generalization of the Schwarz lemmaTheorem (Ahlfors-Schwarz lemma, special case)Let DR be the disc of radius R, with hyperbolic metric λR . For anymetric ρ on DR , such that the curvature Kρ (z) 4 for all z,ρ(z) λR (z)for all z.Eric SchippersHyperbolic metricOctober 15, 201534 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaAhlfors’ generalization of the Schwarz lemmaTheorem (Ahlfors-Schwarz lemma, special case)Let DR be the disc of radius R, with hyperbolic metric λR . For anymetric ρ on DR , such that the curvature Kρ (z) 4 for all z,ρ(z) λR (z)for all z.It says:The hyperbolic metric is maximal, among metrics with boundednegative curvatureIn particular, if f : DR Ω, and λR is the hyperbolic metric on DR ,and λΩ is the hyperbolic metric on Ω, then f λΩ λR .Eric SchippersHyperbolic metricOctober 15, 201534 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaSchwarz lemma is special caseLet f : D D and λ(z) 1/(1 z 2 ) be the hyperbolic metric.Eric SchippersHyperbolic metricOctober 15, 201535 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaSchwarz lemma is special caseLet f : D D and λ(z) 1/(1 z 2 ) be the hyperbolic metric.The curvature of f λ equals 4 since curvature is invariant underpull-back.Eric SchippersHyperbolic metricOctober 15, 201535 / 41

Complex analysisAhlfors’ generalization of the Schwarz lemmaSchwarz lemma is special caseLet f : D D and λ(z) 1/(1 z 2 ) be the hyperbolic metric.The curvature of f λ equals 4 since curvature is invariant underpull-back.So by the Ahlfors-Schwarz lemmaf λ λso f 0 (z) 1 .21 f (z) 1 z 2Eric SchippersHyperbolic metricOctober 15, 201535 / 41

Complex analysisApplications of the Ahlfors-Schwarz lemmaLiouville’s theoremTheoremLet f : C C be an analytic function such that f (z) M for all z C.Then f is constant.Eric SchippersHyperbolic metricOctober 15, 201536 / 41

Complex analysisApplications of the Ahlfors-Schwarz lemmaLiouville’s theoremTheoremLet f : C C be an analytic function such that f (z) M for all z C.Then f is constant.Liouville’s theorem can be interpreted as a limiting case of Schwarzlemma.Eric SchippersHyperbolic metricOctober 15, 201536 / 41

Complex analysisApplications of the Ahlfors-Schwarz lemmaProof of Liouville’s theorem using the Schwarz lemmaMinda, Schober 1983.The hyperbolic metric on the disc of radius R isλR (z) R.R 2 z 2For any R, f maps DR into DM .Eric SchippersHyperbolic metricOctober 15, 201537 / 41

Complex analysisApplications of the Ahlfors-Schwarz lemmaProof of Liouville’s theorem using the Schwarz lemmaMinda, Schober 1983.The hyperbolic metric on the disc of radius R isλR (z) R.R 2 z 2For any R, f maps DR into DM .By the Ahlfors-Schwarz lemma, for any R,f λM λRsoM f 0 (z) R 222M f (z) R z 2for any fixed z.Eric SchippersHyperbolic metricOctober 15, 201537 / 41

Complex analysisApplications of the Ahlfors-Schwarz lemmaProof of Liouville’s theorem using the Schwarz lemmaMinda, Schober 1983.The hyperbolic metric on the disc of radius R isλR (z) R.R 2 z 2For any R, f maps DR into DM .By the Ahlfors-Schwarz lemma, for any R,f λM λRsoM f 0 (z) R 222M f (z) R z 2for any fixed z.Letting R , we get that f 0 (z) 0 for any z. So f c.Eric SchippersHyperbolic metricOctober 15, 201537 / 41

Complex analysisApplications of the Ahlfors-Schwarz lemmaThe Little Picard TheoremTheorem (Little Picard Theorem)Let f : C C be an analytic mapping, whose image omits at least twopoints. Then f is constant.Eric SchippersHyperbolic metricOctober 15, 201538 / 41

Complex analysisApplications of the Ahlfors-Schwarz lemmaThe Little Picard TheoremTheorem (Little Picard Theorem)Let f : C C be an analytic mapping, whose image omits at least twopoints. Then f is constant.The Little Picard theorem is really a case of the (Ahlfors-)Schwarzlemma in disguise.Eric SchippersHyperbolic metricOctober 15, 201538 / 41

Complex analysisApplications of the Ahlfors-Schwarz lemmaproof of the Little Picard TheoremProof.Let p, q be the points omitted from the image of f . Let σ be thehyperbolic metric on C\{p, q} (uniformization theorem!)Eric SchippersHyperbolic metricOctober 15, 201539 / 41

Complex analysisApplications of the Ahlfors-Schwarz lemmaproof of the Little Picard TheoremProof.Let p, q be the points omitted from the image of f . Let σ be thehyperbolic metric on C\{p, q} (uniformization theorem!)For any R, f maps DR into C\{p, q}. Since f σ has curvature 4, wemay apply the Ahlfors-Schwarz lemma:Eric SchippersHyperbolic metricOctober 15, 201539 / 41

Complex analysisApplications of the Ahlfors-Schwarz lemmaproof of the Little Picard TheoremProof.Let p, q be the points omitted from the image of f . Let σ be thehyperbolic metric on C\{p, q} (uniformization theorem!)For any R, f maps DR into C\{p, q}. Since f σ has curvature 4, wemay apply the Ahlfors-Schwarz lemma:σ(f (z)) f 0 (z) f σ λR (z) Eric SchippersHyperbolic metricR2R. z 2October 15, 201539 / 41

Complex analysisApplications of the Ahlfors-Schwarz lemmaproof of the Little Picard TheoremProof.Let p, q be the points omitted from the image of f . Let σ be thehyperbolic metric on C\{p, q} (uniformization theorem!)For any R, f maps DR into C\{p, q}. Since f σ has curvature 4, wemay apply the Ahlfors-Schwarz lemma:σ(f (z)) f 0 (z) f σ λR (z) R2R. z 2Letting R (for any fixed z), we see that σ(f (z)) f 0 (z) 0.Eric SchippersHyperbolic metricOctober 15, 201539 / 41

Complex analysisApplications of the Ahlfors-Schwarz lemmaproof of the Little Picard TheoremProof.Let p, q be the points omitted from the image of f . Let σ be thehyperbolic metric on C\{p, q} (uniformization theorem!)For any R, f maps DR into C\{p, q}. Since f σ has curvature 4, wemay apply the Ahlfors-Schwarz lemma:σ(f (z)) f 0 (z) f σ λR (z) R2R. z 2Letting R (for any fixed z), we see that σ(f (z)) f 0 (z) 0.But σ(f (z)) 6 0, so f 0 (z) 0 (for any z)! So f is constant.Eric SchippersHyperbolic metricOctober 15, 201539 / 41

Complex analysisApplications of the Ahlfors-Schwarz lemmaproof of the Little Picard TheoremProof.Let p, q be the points omitted from the image of f . Let σ be thehyperbolic metric on C\{p, q} (uniformization theorem!)For any R, f maps DR into C\{p, q}. Since f σ has curvature 4, wemay apply the Ahlfors-Schwarz lemma:σ(f (z)) f 0 (z) f σ λR (z) R2R. z 2Letting R (for any fixed z), we see that σ(f (z)) f 0 (z) 0.But σ(f (z)) 6 0, so f 0 (z) 0 (for any z)! So f is constant.This approach due to Minda and Schober (1983). Actually this is avariation on the classical approach. They also give an elementaryproof without using the uniformization theorem.Eric SchippersHyperbolic metricOctober 15, 201539 / 41

ConclusionSummary of hyperbolic complex analysis theoremsThe Schwarz lemma really says that analytic maps from D to Dare hyperbolic contractions.Liouville’s theorem is really a limiting case of the Schwarz lemma.The Little Picard Theorem is really a limiting case of the Schwarzlemma.Actually, any holomorphic map between hyperbolic Riemannsurfaces is a hyperbolic contraction.The hyperbolic metric is central to complex analysis.Eric SchippersHyperbolic metricOctober 15, 201540 / 41

ReferencesSome References1D. Minda and G. Schober, Another elementary approach to thetheorems of Landau, Montel, Picard and Schottky. ComplexVariables 2 (1983) 157–164.2S. Krantz Complex analysis: the geometric viewpoint. CarusMathematical Monographs 23 (1990).3D. Kraus and O. Roth. Conformal metrics. arXiv:0805.2235v1(2008).4L. Ahlfors, An extension of Schwarz’ lemma. Transactions of theAmerican Mathematical Society 43 (1938) 359–364.Upcoming book: D. Minda (Cincinnati) and A. Beardon (Cambridge),The hyperbolic metric in complex analysis. Springer.Eric SchippersHyperbolic metricOctober 15, 201541 / 41

Geometry of conformal metrics The hyperbolic metric on the disc hyperbolic distance Definition The hyperbolic distance between two points z and w is d(z;w) inf Z jdzj 1 j zj2: Definition A geodesic segment between two points is a curve which attains the minimum distance. Warning: This is not the usual definition, but in the case of the