DOI: 10.2478/s11533-007-0013-5 Research article CEJM 5(3) 2007 551–580 Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues∗ Samuel Krushkal Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel and Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA Received 31 July 2006 ; accepted 3 April, 2007 Abstract: The Grunsky and Teichmuller¨ norms κ(f)andk(f) of a holomorphic univalent function f in a finitely connected domain D ∞with quasiconformal extension to C are related by κ(f) ≤ k(f). In 1985, Jurgen¨ Moser conjectured that any univalent function in the disk Δ∗ = {z : |z| > 1} can be approximated locally uniformly by functions with κ(f) <k(f). This conjecture has been recently proved by R. Kuhnau¨ and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kuhnau¨ problem on an exact lower bound in the inverse inequality estimating k(f)byκ(f), and in the related Ahlfors inequality. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: quasiconformal, univalent function, Grunsky coefficient inequalities, universal Teichmuller¨ space, subharmonic function, Strebel’s point, Kobayashi metric, generalized Gaussian curvature, holomorphic curvature, Fredholm eigenvalues. MSC (2000): 30C35, 30C62, 32G15,30F60, 32F45, 53A35 1 Introduction and main results 1.1 Grunsky inequalities and Moser’s conjecture The classical Grunsky theorem states that a holomorphic function f(z)=z+const +O(z−1) in a neighborhood U0 of z = ∞ can be extended to a univalent holomorphic function on ∗ To Reiner Kuhnau¨ on his 70th birthday 552 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 the disk Δ∗ = {z ∈ C = C ∪{∞}: |z| > 1} if and only if its Grunsky coefficients αmn satisfy the inequalities ∞ √ mn αmnxmxn ≤ 1, (1.1) m,n=1 where αmn are defined by ∞ f(z) − f(ζ) log = − α z−mζ−n, (z, ζ) ∈ (Δ∗)2, (1.2) z − ζ mn m,n=1 ∞ 2 2 2 2 x =(xn) runs over the unit sphere S(l )oftheHilbertspacel with x = |xn| ,and 1 the principal branch of the logarithmic function is chosen (cf. [1]). The quantity ∞ √ 2 κ(f):=sup mn αmnxmxn : x =(xn) ∈ S(l ) (1.3) m,n=1 is called the Grunsky constant (or Grunsky norm)off. Let Σ denote the collection of all univalent holomorphic functions −1 ∗ f(z)=z + b0 + b1z + ···:Δ → C \{0}, (1.4) and let Σ(k) be its subset of the functions withk-quasiconformal extensions to the unit {| | } 0 disk Δ = z < 1 so that f(0) = 0. Put Σ = k Σ(k). This collection closely relates to universal Teichmuller¨ space T modelled as a bounded domain in the Banach space B of holomorphic functions in Δ∗ with norm 2 2 ϕ B =sup(|z| − 1) |ϕ(z)|. (1.5) Δ∗ All ϕ ∈ B can be regarded as the Schwarzian derivatives 2 Sf =(f /f ) − (f /f ) /2 of locally univalent holomorphic functions in Δ∗.ThepointsofT represent the functions f ∈ Σ0 whose minimal dilatation μ μ ∗ k(f):=inf{k(w )= μ ∞ : w |∂Δ = f} determines the Teichmuller¨ metric on T.Herewμ denotes a quasiconformal homeomor- phism of C with the Beltrami coefficient μ(z)=∂zw/∂zw, (1.6) and μ ∞ = ess supC |μ(z)| < 1. (1.7) S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 553 Grunsky’s theorem was strengthened for the functions with quasiconformal extensions by several authors. The basic results obtained by Kuhnau,¨ Pommerenke and Zhuravlev are as follows: for any f ∈ Σ0, we have the inequality κ(f) ≤ k(f); (1.8) on the other hand, if a function f ∈ Σ satisfies the inequality κ(f) <kwith some constant k<1, then f has a quasiconformal extension to C with a dilatation k1 = k1(k) ≥ k (see [2–4]and[5], pp. 82–84). An explicit bound k1(k) is given in [6] (see Section 5). A crucial point is that for a generic function f ∈ Σ0,wehavein(1.8) the strict inequality κ(f) <k(f) (1.9) (see, e.g., [7–10]). The functions, for which equality holds in (1.8), play an important role in applications of the Grunsky inequality technique. A characterization of such functions is presented below. In 1985, Jurgen¨ Moser conjectured that the set of functions with κ(f)=k(f) (1.10) is rather sparse in Σ0 so that any function f ∈ Σ is approximated by functions satisfying (1.9) uniformly on compact sets in Δ∗. This conjecture was recently proved in [8]. A related conjecture, posed in [8] and still remaining open, states that f ∈ Σ0 sat- isfying (1.9) cannot be the limit functions of locally uniformly convergent sequences 0 {fn}⊂Σ with κ(fn)=k(fn). This was proved in [11] under the assumption that approximating maps fn are asymp- totically conformal on the unit circle S1 = ∂Δ∗. 1.2 Main theorem Uniform convergence on compact sets is natural and sufficient in many problems of ge- ometric complex analysis. In applications of Schwarzian derivatives, especially to Teich- muller¨ spaces, one has to use the strong topology defined by the norm (1.5). A question is: How sparse is the set of derivatives ϕ = Sf in T representing the maps with the property (1.10)? Our first main result answers this question strengthening Moser’s conjecture. 0 Theorem 1.1. The set of points ϕ = Sf , which represent the maps f ∈ Σ with κ(f) <k(f), is open and dense in the space T. Openness follows from continuity of both quantities κ(f)andk(f) as functions of the Schwarzian derivatives Sf on T (cf. [12]). The main part of the proof concerns the density. The proof involves the density of Strebel points in T and relies on curvature properties of certain Finsler metrics on this space. 554 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 1.3 Application to Fredholm eigenvalues The Fredholm eigenvalues ρn of a smooth closed Jordan curve L ⊂ C are the eigenvalues of its double-layer potential, i.e., of the integral equation ρ ∂ 1 u(z)+ u(ζ) log dsζ = h(z), π ∂nζ |ζ − z| L which often appears in applications (see e.g. [5, 13–18] and the references cited there). These values are intrinsically connected with the Grunsky coefficients of the corre- sponding conformal maps. This is qualitatively expressed by the remarkable Kuhnau-¨ Schiffer theorem on reciprocity of κ(f) to the least positive Fredholm eigenvalue ρL.It is defined for any oriented closed Jordan curve L ⊂ C by 1 |D (u) −D ∗ (u)| =sup G G , ρL DG(u)+DG∗ (u) where G and G∗ are, respectively, the interior and exterior of L; D denotes the Dirichlet integral, and the supremum is taken over all functions u continuous on C and harmonic on G ∪ G∗ (cf. [10, 16]). In general, there is only a rough estimate for ρL by Ahlfors’ inequality 1 ≤ qL, (1.11) ρL where qL is the minimal dilatation of quasiconformal reflections across L, (that is, of the orientation reversing quasiconformal homeomorphisms of C preserving L point-wise); see, e.g., [15, 19, 20]. As a consequence of Theorem 1.1,wehave Theorem 1.2. The set of quasiconformal curves L, for which Ahlfors’ inequality (1.11) is satisfied in the strict form 1/ρL <qL, is open and dense in the strongest topology determined by the norm (1.5). Proof. Since all quantities in (1.11) are invariant under the action of the M¨obius group PSL(2, C)/ ± 1, it suffices to exploit quasiconformal homeomorphisms f of the sphere C 1 ∗ carrying the unit circle S = ∂Δ onto L whose Beltrami coefficients μf (z)=∂z¯f/∂zf have support in the unit disk Δ and which are normalized via (1.4), i.e., with restrictions ∗ 0 μ f|Δ ∈ Σ . Then the reflection coefficient qL equals the minimal dilatation k(w )= μ ∞ of quasiconformal extensions wμ of f|Δ∗ to C,andTheorem1.2 immediately follows from Theorem 1.1. In the last section we solve Kuhnau’s¨ problem related to comparing the Teichmuller¨ and Grunsky norms and establish the sharp lower bound in the inverse inequalities esti- mating k(f)byκ(f)orρL. This important problem has been open since 1981. S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 555 2 Preliminaries We briefly present here certain underlying results needed for the proof of Theorem 1.1. The exposition is adapted to our special case. 2.1 Frame maps and Strebel points For a map f μ ∈ Σ0,wedenoteby[f μ] its equivalence class containing the maps f ν ∈ Σ0 which coincide with f μ on the unit circle S1 (and hence on Δ∗). μ0 Let f0 := f be an extremal representative of its class [f0] with dilatation μ μ 1 1 k(f0)= μ0 ∞ =inf{k(f ):f |S = f0|S } = k, and assume that there exists in this class a quasiconformal map f1 whose Beltrami coef- ficient μf1 satisfies the strong inequality | | ess supAr μf1 (z) <k in some annulus Ar := {z : r<|z| < 1}.Thenf1 is called a frame map for the class [f0] and the corresponding point of the space T is called a Strebel point. The following two results are fundamental in the theory of extremal quasiconformal maps and Teichmuller¨ spaces. Proposition 2.1. (cf. [21]) If a class [f] has a frame map, then the extremal map f0 in this class is unique and either conformal or a Teichmuller¨ map with Beltrami coefficient μ0 = k|ψ0|/ψ0 on Δ, defined by an integrable holomorphic quadratic differential ψ on Δ and a constant k ∈ (0, 1).