Complex geodesics and variational calculus for univalent functions

Theorem 1.1. The Carath´eodory metric of the space eT coincides with its Kobayashi metric, hence all invariant non-expanding metrics on eT are equal its Teichm¨uller metric, and c eT (X1, X2) = d eT (X1, X2) = τ eT (X1, X2) = inf{dD(h−1(ϕ), h−1(ψ)) : h ∈Hol(D, eT)}, where dD denotes the hyperbolic metric of the unit disk of curvature −4. Similarly, the infinitesimal forms of these metrics coincide with the Finsler metric FeT(ϕ, v) generating τeT and have holomorphic sectional curvature −4. This w

Theorem 1.2. (i) Any two points of the space eT can be joined by a complex geodesic. The geodesic joining a Strebel point with the base point is unique and defines the corresponding Teichm¨uller extremal disk. (ii) For any point ϕ ∈eT and any nonzero tangent vector v at this point, there exists at least one complex geodesic h : D →eT such that h(0) = ϕ and h′(0) is collinear to v. 1.3. The coincidence of invariant metrics and existence of complex geodesics was known only for convex domains in dua

Theorem 1.1 implies, together with the definition of complex geodesics, that these geodesics in eT are the Teichm¨uller geodesic disks in this space. Such disks are uniquely determined for Strebel points; on the other hand, Tanigawa constructed in [38] the extremal Beltrami coefficients µ0 with nonconstant |µ0(z)| < ∥µ0∥∞on a set of positive measure for which there exist infinitely many distinct geodesic segments in the universal Teichm¨uller space T joining the points φT(0) and φT(µ0). All these se

Lemma 2.1. [16], [21] For all f ∈Σ0, κ(f) ≤k k + αD(f) 1 + αD(f)k, k = k(f), and κ(f) < k unless αD(f) = ∥µ0∥∞, (2.3) and the last equality is equivalent to κ(f) = k(f). Moreover, for small ∥µ∥∞, κr(f) = sup |⟨µ, ψ⟩D| + O(∥µ∥2 ∞), ∥µ∥∞→0,

Lemma 2.2. In the case b1 ̸= 0, there holds for sufficiently small |t| ≤r0(f) the equality κ(ft) = k(ft). Indeed, due to [27], for small |t| the extremal quasiconformal extension of ft to D is defined by a nonvanishing holomorphic quadratic differential. Generically, r0(f) < 1, which is connected with critical points of the homotopy disk. Note that Sf(z) = −6b1z−4 + O(z−5), so b1 = 0 only for f with lim z→∞z4Sf(z) = −6b1 = 0. (2.6) It suffices to prove the theorem for the set of Sf ∈T such that b1 ̸=

Lemma 2.3. Let D be a simply connected domain on the Riemann sphere bC. Assume that there are a set E of positive two-dimensional Lebesgue measure and a finite number of points z1, z2, ..., zm distinguished in D. Let α1, α2, ..., αm be non-negative integers assigned to z1, z2, ..., zm, respectively, so that αj = 0 if zj ∈E. Then, for a sufficiently small ε0 > 0 and ε ∈(0, ε0), and for any given collection of numbers wsj, s = 0, 1, ..., αj, j = 1, 2, ..., m which satisfy the conditions w0j ∈D, |w0j

Lemma 2.4. Let Y be reflexive and let Ωbe a bounded set in Hol(G, Y ). Then any sequence from Ωcontains a subsequence which weakly converges to a holomorphic map from G to Y . Hence, the compactness (strong or weak) is actually required only for the image of G in Y under a given family of maps. In our case Y = C, and for each x ∈eT its orbit h(x) is located in the unit disk which is compact. This implies that for any point x0 from eT there exists a holomorphic map h0 : eT →D with h0(0) = 0 and dD

Lemma 2.5. [29] If a function u : Ω→[−∞, +∞) is upper semicontinuous in a domain Ω⊂C and its generalized Laplacian satisfies the inequality ∆u(z) ≥Ku(z) with some positive constant K at any point z ∈Ω, where u(z) > −∞, and if lim sup z→ζ u(z) ≤0 for all ζ ∈∂Ω, then either u(z) < 0 for all z ∈Ωor else u(z) = 0 for all z ∈Ω. It is applied to the ratio u = log(λC/λK) = log λC −log λK. One needs to consider only the points Sft with |t| > r0. In view of the upper semicontinuity of both metrics the set

Lemma 2.5 that λC = λK on U0. Continuing in a similar way, one extends this equality first to all noncritical points of the disk D(Sf) and then by continuity of metrics to the whole homotopy disk. This also provides the equality of the global distances cT and dT on this disk. Now, by applying the infinitesimal analog of Montel’s normality theorem and Lemma 2.4, one gets, letting t →1, cT(0, Sf) = dT(0, Sf). (2.9) The case of two arbitrary points ϕ1, ϕ2 ∈T is reduced to (2.9) in a standard way usin

Theorem 3.1. (i) For any functional J of type (3.3) whose range domain J(Σ0(D∗)) has more than two boundary points, there exists a number k0(J) > 0 such that for all k ≤k0(J), we have the sharp bound max ∥µ∥≤k |J(f µ) −J(id)| ≤max |t|=k |J(f t|ψ0|/ψ0) −J(id)|; (3.4)

Lemma 3.2. Let W be a non-trivial closed (complex) subspace of V , and let W ′ be the closed subspace W ′ = {w ∈V : A(v, w) = 0 for all v ∈W \ {0}. There is a projection P of norm 1 from V onto W if and only if W ′ is a complementary subspace to W, that is W ⊕W ′ = V . Further, if P exists, it is unique and its kernel is W ′. The next lemma is a straightforward modification of the corresponding lemmas of Royden [33] and of Earle and Kra [5]. Let a1, a2, . . . , an be distinguished points of a dom

Lemma 3.3. The function h(t) is differentiable near t = 0, and h′(0) = Re ZZ D ψ(z)ϕ(z) ϕ(z)dxdy. (3.6) Moreover, if αj ≤2βj+1 for all j = 1, . . . , n, then the second derivative h′′(0) exists. If αj > 2βj+1 for some j, then h(t) = h(0) + th′(0) + n X 1 cjδj(t) + o(max j

Lemma 3.4. For sufficiently small k ≤k0(J), the extremal Beltrami coefficient µf0 is orthogonal to all functions (3.11), i.e., ⟨µf0, ψp⟩D = 0. (3.12)

Theorem 3.1 provides new various explicit estimates controlling the distortion in both conformal and quasiconformal domains simultaneously. A similar theorem is valid also for univalent functions on bounded quasidisks D, for example, for the canonical class Sk(D) of univalent functions in D normalized by f(z) = z + c2z2 + . . . near the origin (provided that z = 0 ∈D) and admitting k-quasiconformal extensions to bC which preserve the infinite point. Earlier only very special results have been est

Corollary 4.1. In any class of univalent functions with κ-quasiconformal extension, no function can be simultaneously extremal for different holomorphic functionals (3.4) unless these functionals have equal 1-jets at the origin. 4.2. Example: the coefficient problem for functions with quasiconformal extensions. We mention here an improvement in estimating the Taylor coefficients. Though the Bieberbach conjecture for the canonical class S of univalent functions f(z) = z + ∞ P 2 anzn in D has already b

Theorem 4.2. [17] For all f ∈Sk(∞) and all k ≤1/(n2 + 1), |an| ≤2κ/(n −1), (4.2) with equality only for the functions fn−1,t(z) = f1,t(zn−1)1/(n−1) = z + 2t n −1zn + . . . , n = 3, 4, . . . ; |t| = k. (4.3) The estimate (4.2) also holds in the classes Sk(1) with the same bound for k. Until now, no estimates have been obtained for arbitrary k < 1, unless n = 2; in the last case, |a2| ≤2k with equality for the function (4.1) when |t| = k. The rigidity provided by Corollary 4.1 yields that the func

Theorem 4.3. For every functional (4.7) and any finite set e of fixed points defined above, there exists a positive number k0(J, e) < 1 such that for all k ≤k0(J, e), we have for any function f ∈Σk(D∗, 1, e) the sharp bound max ∥µ∥≤k |J(f µ)| = max |t|=κ |J(f t|ψe|/ψe)| = dκ + O(κ2) (4.9) with uniformly bounded ratio O(κ2)/κ2. Here κ = κ(k) < k is the bound for Teichm¨uller norms of f ∈Σk(D∗, 1, e) (i.e, regarding these f as the functions of Σ(D∗, 1) with omitted restrictions (4.8)), ψe = ψ0 + m X

Proposition 4.4. For 0 < k < k0(J, ζ0) ≤1, max{|J(f µ) : f µ ∈Σ(D∗, 1, e), ∥µ∥≤k} < dκ, (4.15) where κ = κ(k) and d is the L1-distance between the differential ψ0 and the line c ζ0 −1 (z −1)(z −ζ0), c ∈C. In contrast to Theorems 3.1 and 4.3, the bound (4.15) is not sharp. Similar results are valid also for the over-normalized functions in bounded quasidisks. We illus- trate those again on the coefficient problem: Find max |an| (n ≥2) for the functions f ∈Sk(∞) leaving a given set e = (e1, . . . , e

Theorem 4.5. For any n ≥2, there is a number kn(e) < 1 such that for k ≤kn and all f ∈Sk(∞), which fix a given set e = (e1, . . . , em), we have the sharp bound max ∥µ∥≤k |an(f µ)| = max |t|=κ |an(f t|ψn|/ψn)| = 2dnκ n −1 + O(κ2), (4.16) where κ(k) is again the bound for Teichm¨uller norms, and similar to (4.10),(4.11), ψn(z) = cz−n−1 + m X 1 ξsρs(z), dn = inf L(e) ∥ψn −ψ∥1.

Theorem 4.6. For small r > 0, the minimal dilatation, on which any functional (4.17) attains its level surface Lr = {|J(f)| = r}, equals κ = r/d + O(r2) with d defined by (4.11). This estimate is sharp. The problem of establishing sharp explicit global distortion estimates for over-normalized maps, even such as Theorem 3.1, is very complicated and remains open. I am thankful to the referee for his comments and suggestions. References [1] L. Bers, Fiber space over Teichm¨uller spaces, Acta Math. 1