Hyperbolicity theory for minimal surfaces in Euclidean spaces
aa r X i v : . [ m a t h . DG ] F e b Schwarz–Pick lemma for harmonic mapswhich are conformal at a point
Franc Forstneriˇc and David Kalaj
Abstract
We obtain a sharp estimate on the norm of the differential of a harmonicmap from the unit disc D in C to the unit ball B n in R n , n ≥ , at any point wherethe map is conformal. In dimension n = 2 this generalizes the classical Schwarz–Pick lemma to harmonic maps D → D which are conformal at the reference point.In dimensions n ≥ it gives the optimal Schwarz–Pick lemma for conformalminimal discs D → B n . Let M denote the restriction of the Kobayashi metricon the complex n -ball to the real n -ball B n . We show that conformal harmonicimmersions M → ( B n , M ) from any hyperbolic open Riemann surface M withits natural Poincaré metric are distance-decreasing, and the extremal maps areprecisely the conformal embeddings of D onto affine discs in B n . Keywords harmonic map, conformal minimal surface, Schwarz–Pick lemma
MSC (2020):
Primary: 30C80, 31A05, 53A10
Date:
20 February 2021
1. Introduction
In this paper, we establish precise estimates of derivatives and the rate of growth ofconformal harmonic maps from hyperbolic open Riemann surfaces into the unit ball B n of R n (see (1.2)) for any n ≥ . In dimensions n ≥ such maps parameterize minimalsurfaces, which are objects of high interest in geometry. More on this in the sequel.One of our main results, Theorem 2.1, gives a sharp upper bound on the operator norm k df z k of the differential of a harmonic map f from the unit disc D = { z ∈ C : | z | < } into the ball B n for n ≥ at any point z ∈ D where the map is conformal. In dimension n = 2 , it gives the following generalization of the classical Schwarz–Pick lemma , due to H.A. Schwarz [23] (1869) with an addition by G. A. Pick [21] (1915).
Theorem 1.1. If f : D → D is a harmonic map which is conformal at a point z ∈ D , thenat this point we have (1.1) k df z k ≤ − | f ( z ) | − | z | , with equality if and only if f is a conformal diffeomorphism of the disc D . The classical Schwarz–Pick lemma gives the same conclusion under the substantiallystronger hypothesis that the map f is conformal at every noncritical point; equivalently,that it is holomorphic or antiholomorphic (see Kobayashi [15, Theorem 1.1] or Dineen[5]). Omitting the conformality assumption at the point z , the estimate (1.1) fails for someharmonic diffeomorphisms of D ; see Example 6.1. F. Forstneriˇc and D. KalajThe conditions in Theorem 1.1 are invariant under precompositions by holomorphicautomorphisms of the disc D , so the proof reduces to the case when the reference pointis z = 0 . On the other hand, postcompositions of harmonic maps D → D by holomorphicautomorphism of D need not be harmonic, so we cannot exchange f (0) and . The standardproof of the classical Schwarz–Pick lemma breaks down at this point.Holomorphic maps without critical points between plane domains are preciselyorientation preserving conformal maps, in particular, conformal harmonic maps. Their mostnatural generalization are conformal harmonic immersions from open Riemann surfacesinto Euclidean spaces R n for n ≥ ; these parameterize minimal surfaces in R n , which arehighly interesting and extensively studied objects. Indeed, a smooth conformal immersion f : M → R n from an open Riemann surface parameterizes a minimal surface in R n if andonly if f is a harmonic map (see Osserman [20] or Duren [6]). We shall consider harmonicmaps from the disc and other open Riemann surfaces into the unit ball(1.2) B n = n x = ( x , . . . , x n ) ∈ R n : | x | = n X k =1 x k < o of R n for any n ≥ . In Theorem 2.1 we provide a precise upper bound on the norm ofthe differential df z of a harmonic map f : D → B n at any point z ∈ D where the map isconformal. The estimate is similar to the one in Theorem 1.1, except that it also involves theangle θ between the position vector f ( z ) ∈ B n and the 2-plane df z ( R ) ⊂ R n . A relatedresult, Theorem 2.2, shows that the worst estimate, which occurs for θ = 0 (i.e., when f ( z ) ∈ df z ( R ) ) is satisfied for all harmonic maps f : D → B n without any conformalityassumption, provided that the operator norm is replaced by the size of the gradient.We then give applications of Theorems 2.1 and 2.2 to conformal minimal surfaces,allowing branch points of harmonic maps. Theorem 2.3 shows that the extremal conformalminimal discs D → B n for any n ≥ are precisely the conformal parameterizations ofaffine discs obtained by intersecting B n with affine -planes. It also provides the preciserate of growth of discs with the centre f (0) = 0 . This result has a natural differential-theoretic formulation, analogous to those in Kobayashi’s theory for holomorphic maps;see Kobayashi [14, 15] for the latter. Let M denote the Riemannian metric on the ball B n obtained by restricting the Kobayashi metric on the complex ball B n C (2.5) to pointsof B n and real tangent vectors (see Definition 2.5). Then, any conformal harmonic map f : M → B n (possibly with branch points) from an open Riemann surface is distancedecreasing with respect to the Poincaré metric on M and the metric M on B n . Furthermore,if the differential df p has the operator norm equal to at some point p ∈ M , or if thedistances are preserved for a pair of distinct points in M , then M is the disc D and f is aconformal diffeomorphism onto an affine disc in B n , having the operator norm at everypoint; see Theorem 2.8. We give several other corollaries on the way.In Section 2 we give precise statements of all main results. The proof of Theorem 2.1is outlined in Section 3, with the main part given by Lemma 4.1. We introduce a new ideainto the subject, connecting it with Lempert’s seminal work [19] from 1981 on complexgeodesics of the Kobayashi metric on bounded convex domains of C n . In Section 5 weprove the distance-decreasing property of conformal harmonic maps from hyperbolic openRiemann surfaces into the ball B n . Theorem 2.2 is proved in Section 6. In Section 7 weapply the new Schwarz–Pick lemma for harmonic maps, given by Theorem 1.1, to estimatethe gradient of a quasiconformal harmonic map of the disc into itself in terms of the secondBeltrami coefficient of the map at the reference point; see Theorem 7.1.chwarz–Pick lemma for harmonic maps which are conformal at a point 3
2. The main results
Given a differentiable map f = ( f , . . . , f n ) : D → R n , we denote by f x and f y its partial derivatives with respect to x and y , where z = x + i y ∈ D . The gradient ∇ f = ( f x , f y ) is an n × matrix representing the differential df . The map f is said to be conformal at z ∈ D if the differential df z preserves angles, which holds if and only if(2.1) | f x ( z ) | = | f y ( z ) | > and f x ( z ) · f y ( z ) = 0 . Here, the dot stands for the Euclidean inner product on R n , and | x | is the Euclidean normof x ∈ R n . We allow maps to have branch points; the estimates that we shall present aretrivially fulfilled at such points. We define |∇ f ( z ) | = | f x ( z ) | + | f y ( z ) | . If f is conformal at z then k df z k = √ − |∇ f ( z ) | = | f x ( z ) | = | f y ( z ) | . The map f = ( f , . . . , f n ) : D → R n is harmonic if and only if every component f k is a harmonicfunction on D , meaning that the Laplacian ∆ f k = ∂ f k ∂x + ∂ f k ∂y vanishes identically.Our first main result is the following. Theorem 2.1.
Let f : D → B n for n ≥ be a harmonic map which is conformal at apoint z ∈ D . Denote by θ ∈ [0 , π/ the angle between the vector f ( z ) and the plane Λ = df z ( R ) ⊂ R n . (When f ( z ) = 0 , the angle will not matter.) Then at this point we have (2.2) k df z k = 1 √ |∇ f ( z ) | ≤ − | f ( z ) | − | z | p − | f ( z ) | sin θ , with equality if and only if f is a conformal diffeomorphism of D onto the affine disc ( f ( z ) + Λ) ∩ B n . Hence, equality in (2.2) at one point z ∈ D implies equality at all points. The proof ofTheorem 2.1 is outlined in Section 3, with the main part given by Lemma 4.1. In dimension n = 2 we necessarily have θ = 0 , so Theorem 1.1 is a special case of Theorem 2.1. Withoutassuming that the map f is conformal at z or that f ( z ) = 0 , the inequality (2.2) fails forsome harmonic diffeomorphisms of the disc as shown in Example 6.1.Note that for a fixed value of | f ( z ) | ∈ [0 , , the maximum of the right hand side of (2.2)over angles θ ∈ [0 , π/ equals √ −| f ( z ) | −| z | and is reached precisely at θ = π/ , i.e, whenthe vector f (0) is orthogonal to Λ = df z ( R ) , unless f ( z ) = 0 in which case it equals −| z | for all θ . We show that this weaker estimate holds for all harmonic maps D → B n without any conformality assumption. Theorem 2.2.
For every harmonic map f : D → B n ( n ≥ we have that (2.3) √ |∇ f ( z ) | ≤ p − | f ( z ) | − | z | , z ∈ D . Equality holds for some z ∈ D if f ( z ) is orthogonal to the -plane Λ = df z ( R ) and f is a conformal diffeomorphism onto the affine disc ( f ( z ) + Λ) ∩ B n . In particular,if f ( z ) = 0 then |∇ f ( z ) | ≤ √ −| z | , with equality if and only if f is a conformaldiffeomorphism onto the linear disc Λ ∩ B n . F. Forstneriˇc and D. KalajThe proof of Theorem 2.2 is given in Section 6. The estimate (2.3) only uses thehypothesis that the L -norm of | f | = P nk =1 f k on the circles {| z | = r } for < r < is bounded by . This clearly holds for maps into the ball; however, we do not knowwhether there are harmonic maps reaching (near) equality in (2.3) whose images are actuallycontained in the ball. In this connection, see the discussion following Theorem 7.1.Theorem 2.2 implies a bound on the area of the image of f . It is classical (see Lawson[18, p. 61]) that for a C map f : D → R n ( n ≥ from a domain D ⊂ C we have Area f ( D ) ≤ Z D |∇ f | dxdy, with equality if and only if f is conformal. (The area is counted with multiplicities.) Hence,if f : D → B n is a harmonic map then (2.3) implies for every < r < that(2.4) Area f ( r D ) ≤ Z | z |
Harmonic maps in the above two theorems need notparameterize minimal surfaces, unless they are conformal at every noncritical point. Forconformal harmonic maps (allowing branch points), Theorem 2.1 gives the optimal estimateon the norm of the differential at every point of the disc, which is the first part of thefollowing theorem. This implies the second part concerning growth of such discs.chwarz–Pick lemma for harmonic maps which are conformal at a point 5
Theorem 2.3 (Schwarz–Pick lemma for conformal minimal discs) . Let f : D → B n for n ≥ be a conformal minimal disc. (a) The inequality (2.2) holds at every point z ∈ D . If equality holds at one point, then itholds at all points, and this is true if and only if f is a conformal diffeomorphism ontoan affine disc in B n , the intersection of B n with an affine plane. (b) If f (0) = 0 then | f ( z ) | ≤ | z | for all z ∈ D . Equality at one point z ∈ D \ { } impliesthat f is a conformal parameterization of a linear disc obtained by intersecting B n witha plane through the origin, and hence equality holds at all points. Theorem 2.3 (b) is proved in Section 5.Let us mention a consequence related to the Schwarz lemma for holomorphic discs (seeRudin [22, Sect. 8.1]). Holomorphic immersions D → C n are a small subset in the space ofconformal harmonic immersion D → R n . The following corollary to Theorem 2.3 showsthat the Kobayashi extremal holomorphic discs in the unit ball(2.5) B n C = n z = ( z , . . . , z n ) ∈ C n : | z | = n X k =1 | z k | < o of the complex Euclidean space C n are precisely those extremal orientation preservingconformal minimal immersions D → B n C which parameterize affine complex discs. Corollary 2.4 (Schwarz–Pick lemma for conformal harmonic discs in a complex ball) . Assume that f : D → B n C is a harmonic map which is conformal at a point z ∈ D . If Λ = df z ( R ) is a complex line in C n , then equality holds in (2.2) for this z if and only if f is a biholomorphic or anti-biholomorphic map onto the affine complex disc ( f ( z )+Λ) ∩ B n C . The minimal metric.
We can interpret Theorem 2.3 as the distance-decreasing property ofconformal harmonic maps D → B n with respect to a certain Riemannian metric M on B n which we now introduce. Recall that | v | denotes the Euclidean length of v ∈ R n . Definition 2.5.
The minimal metric M on B n is given by(2.6) M ( x , v ) = p − | x | sin φ − | x | | v | , x ∈ B n , v ∈ R n , where φ ∈ [0 , π/ is the angle between the vector x and the line R v ⊂ R n . Equivalently,(2.7) M ( x , v ) = (1 − | x | ) | v | + | x · v | (1 − | x | ) = | v | − | x | + | x · v | (1 − | x | ) . We have that | v | p − | x | ≤ M ( x , v ) ≤ | v | − | x | , with the upper bound reached for the angle φ = 0 and the lower bound for φ = π/ . Therelevance of this metric is seen by observing that the inequality (2.2) can be rewritten as(2.8) p − | f ( z ) | sin θ − | f ( z ) | | df z ( ξ ) | ≤ | ξ | − | z | , ξ ∈ T z D = R , where θ ∈ [0 , π/ is the angle between f ( z ) ∈ B n and the 2-plane Λ = df z ( R ) . Sincethe angle φ between the real line f ( z ) R ⊂ R n and the vector df z ( ξ ) ∈ Λ clearly satisfies φ ≥ θ , we have − sin φ ≤ − sin θ and hence the left hand side of (2.8) is bigger or equalto M ( f ( z ) , df z ( ξ )) . This, and the addition concerning equality in (2.2), gives the following. F. Forstneriˇc and D. Kalaj Corollary 2.6. If f : D → B n is a conformal harmonic immersion then (2.9) M (cid:0) f ( z ) , df z ( ξ ) (cid:1) ≤ | ξ | − | z | , z ∈ D , ξ ∈ R . Thus, f is a distance decreasing map from the disc D with the Poincaré metric | dz | −| z | to theball B n with the minimal metric M given by (2.6) . If equality holds in (2.9) for some z ∈ D and ξ ∈ R \ { } , then f is a conformal parameterization of an affine disc in B n . Remark 2.7.
The minimal metric M given by (2.7) is the restriction of the Kobayashimetric on the unit ball B n C ⊂ C n to points x ∈ B n = B n C ∩ R n and tangent vectors in T x R n ∼ = R n . It also equals / √ n + 1 times the Bergman metric on B n C restricted to B n and real tangent vectors; see Krantz [17, Proposition 1.4.22]. (On the ball of C n , mostholomorphically invariant metrics coincide up to scalar factors.) The metric M is notconformally equivalent to the Euclidean metric on B n , and it does not coincide with thePoincaré metric given by | v | −| x | . More precisely, it coincides with P in the radial directionparallel to the base point x ∈ B n , but it is strictly smaller in the direction perpendicular to x as seen from (2.7). (cid:3) We now consider conformal minimal surfaces M → B n parameterized by arbitraryhyperbolic open Riemann surface M , i.e., one whose universal covering space is the disc D . (Nonhyperbolic Riemann surfaces do not admit any nonconstant harmonic maps tothe ball, so they need not be considered.) Denote by Aut( D ) the group of holomorphicautomorphisms of D . Let h : D → M be a universal holomorphic covering map. Thegroup Γ ⊂ Aut( D ) of deck transformations of h acts without fixed points and totallydiscontinuously on D , and the action is transitive on every fibre of h . Since elements of Aut( D ) are isometries of the Poincaré metric | dz | −| z | (see [15]), there is a unique Kählermetric P M on M , called the Poincaré metric on M , such that the projection h is a localisometry. This metric agrees with the Kobayashi metric of M as seen by noting thatevery holomorphic map g : D → M lifts to a holomorphic map ˜ g : D → D such that h ◦ ˜ g = g ; furthermore, ˜ g is uniquely determined by the choice of the value ˜ g (0) in the fibre h − ( g (0)) ⊂ D . In particular, k dg k reaches the maximal value if and only if g : D → M is a holomorphic covering. It is well known and easily seen that P M is a complete Kählermetric of the same constant Gaussian curvature as the Poincaré metric P D ( z, ξ ) = | ξ | −| z | ,which is − . (See [15, p. 48, Example 2].) The following result generalizes Corollary 2.6. Theorem 2.8 (Distance decreasing property of conformal harmonic maps) . Let M be ahyperbolic open Riemann surface endowed with the Poincaré metric P M . Then, everyconformal harmonic map f : M → B n ( n ≥ satisfies (2.10) M (cid:0) f ( p ) , df p ( ξ ) (cid:1) ≤ P M ( p, ξ ) , p ∈ M, ξ ∈ T p M. If equality holds for some point p ∈ M and vector = ξ ∈ T p M , then M = D and f : D → B n is a conformal diffeomorphism onto an affine disc in B n .Proof. Choose a holomorphic covering map h : D → M and a point z ∈ D with h ( z ) = p . The conformal harmonic map ˜ f = f ◦ h : D → B n satisfies ˜ f ( z ) = f ( p ) and d ˜ f z = df p ◦ dh z . Let η ∈ R be such that dh z ( η ) = ξ . Then, P M ( p, ξ ) = P D ( z, η ) bythe definition of the metric P M , and d ˜ f z ( η ) = df p ( ξ ) . From (2.9) it follows that M (cid:0) f ( p ) , df p ( ξ ) (cid:1) = M (cid:0) ˜ f ( z ) , d ˜ f z ( η ) (cid:1) ≤ | d ˜ f z ( η ) | − | ˜ f ( z ) | = | df p ( ξ ) | − | f ( p ) | , chwarz–Pick lemma for harmonic maps which are conformal at a point 7which gives (2.10). If ξ = 0 and equality holds, then we see from Corollary 2.6 that ˜ f = f ◦ h : D → B n is a conformal diffeomorphism onto an affine disc in B n , from whichit clearly follows that h : D → M is a biholomorphism. (cid:3) Corollary 2.6 and Theorem 2.6 obviously imply the distance-decreasing property ofconformal harmonic maps f : M → B n for the distance functions on the Riemann surface M and the ball B n , obtained by taking the infimum of lengths of paths connecting a givenpair of points in the respective Riemannian metrics on the two manifold. On M , this is thePoincaré distance function. On the disc with the Poincaré metric P D = | dz | −| z | , it equals(2.11) dist( z, w ) = 12 log (cid:18) | − zw | + | z − w || − zw | − | z − w | (cid:19) , z, w ∈ D . The distance function on B n determined by the minimal metric M (2.6) coincides up toa scalar factor √ n + 1 with the restriction to B n of the Bergman distance function on thecomplex ball B n C , or equivalently with the restriction to B n of the Kobayashi distance on B n C . An explicit formula for z , w ∈ B n C can be found in [17, p. 437]:(2.12) dist( z , w ) = 12 log | − z · w | + p | z − w | + | z · w | − | z | | w | | − z · w | − p | z − w | + | z · w | − | z | | w | ! . Here, z · w = P k z k w k . If the distances agree for a pair of distinct points in M and theirimages in B n , we conclude that the differential df p has norm at some point of M , andhence M = D and f is a conformal diffeomorphism onto an affine disc in B n . Asymptotic behaviour of the minimal metric near the boundary.
When the point x ∈ B n approaches the boundary b B n , the formula (2.7) gives an asymptotic rate of growthof the minimal metric, which is analogous to the well-known estimates of invariant metricsin the complex case; see Graham [10] for the Kobayashi metric and Fefferman [7] for theBergman metric on strongly pseudoconvex domains. (There is an extensive literature onthis subject.) Writing a vector v ∈ R n as v = v T + v N , where v N is parallel to x and v T is orthogonal to x , and letting δ ( x ) = 1 − | x | denote the distance of x to b B n , we have(2.13) M ( x , v ) ≈ | v T | δ + | v N | δ as δ → . One can introduce the minimal metric on an arbitrary domain D ⊂ R n ( n ≥ as theFinsler pseudometric on the Grassmann manifold D × G ( R n ) of all pairs ( x , Λ) , where x ∈ D and Λ ⊂ R n is a linear -plane, given by F D ( x , Λ) = inf (cid:8) / k df k : f ∈ CMI( D , D ) , f (0) = x , df ( R ) = Λ (cid:9) . Here, k df k is the operator norm in the Euclidean metric on R n and CMI( D , D ) denotesthe space of conformal minimal immersions D → D . If D is a domain in C n , Λ ⊂ C n is acomplex line, and we only use holomorphic maps in the above definition, then F D ( x , Λ) isthe Kobayashi length in D of any vector v ∈ Λ having Euclidean length . On any boundedstrongly convex domain D ⊂ R n with C boundary, (2.13) gives asymptotic estimates of F ( x , Λ) by applying the comparison principle with inscribed and circumscribed balls at theboundary point q ∈ bD which is closest to the centre x of the disc; note that such q is uniqueif x is close enough bD . However, it is not clear how to do this for nonconvex domains since,unlike in the holomorphic case for the Kobayashi metric, changes of coordinates other thanrigid motions of R n are not allowed. F. Forstneriˇc and D. KalajThe class of domains in real Euclidean spaces R n ( n ≥ , which plays a similar rolefor conformal minimal surfaces as (strongly) pseudoconvex domains in complex Euclideanspaces C n play in complex analysis, is the class of (strongly) minimally convex domains ;see Alarcón et al. [1]. A domain D ⊂ R with C boundary is (strongly) minimally convexif the mean curvature of the boundary bD is nonnegative (resp. positive) from the interiorside at every point q ∈ bD ; such domains are also called mean-convex . In particular, bD need not be locally convex. The analysis of asymptotic boundary estimates would requirea Schwarz-Pick lemma for conformal minimal discs in model domains of the local form z > ax + by with a + b > at (0 , , ∈ R , along with a generalization of thelocalization principle of Forstneriˇc and Rosay [8] to this situation.
3. Towards the proof of Theorem 2.1
We begin by an explicit description of extremal discs mentioned in Theorem 2.1. Usingthese discs, we then indicate how Theorem 2.1 reduces to Lemma 4.1 proved in thefollowing section. Theorems 2.2 and 2.3 are proved in Sections 6 and 5, respectively.The first observation is that it suffices to prove Theorem 2.1 for z = 0 . Indeed, with f and z as in the theorem, let φ z ∈ Aut( D ) be a holomorphic automorphism such that φ z (0) = z . The harmonic map g = f ◦ φ z : D → B n is then conformal at the origin. Since | φ ′ z (0) | = 1 − | z | , (2.2) follows from the same estimate for the map g applied at z = 0 .The second observation is that the hypotheses and statement of Theorem 2.1 are invariantunder postcomposition of maps D → B n by orthogonal rotations of R n , i.e., elements of theorthogonal group O n . Fix a point q ∈ B n and a linear -plane ∈ Λ ⊂ R n , and considerthe affine disc Σ = ( q + Λ) ∩ B n . Let us identify conformal parameterizations D → Σ sending to q . Let p ∈ Σ be the closest point to the origin. If n = 2 then p = 0 and Σ = D . Suppose now that n ≥ . By an orthogonal rotation we may assume that(3.1) p = (0 , , p, . . . , and Σ = (cid:8) ( x, y, p, , . . . ,
0) : x + y < − p (cid:9) . Let q = ( b , b , p, , . . . , ∈ Σ , and let θ denote the angle between q and Σ . Set(3.2) c = p − p = q − | q | sin θ, a = b + i b c ∈ D , | a | = | q | cos θc . We orient Σ by the tangent vectors ∂ x , ∂ y in the parameterization (3.1). Every orientationpreserving conformal parameterization f : D → Σ with f (0) = q is then of the form(3.3) f ( z ) = (cid:18) c ℜ e i t z + a ae i t z , c ℑ e i t z + a ae i t z , p, , . . . , (cid:19) , z ∈ D for some t ∈ R . (Here, ℜ and ℑ stand for the real and imaginary part of a complexnumber. If n = 2 then p = 0 , c = 1 , and the same holds if we drop all coordinates exceptthe first two. Orientation reversing conformal parameterizations are obtained by replacing z = x + i y with ¯ z = x − i y . By a rotation in the ( x, y ) -plane we may further assume that b = 0 and f (0) = ( b , , p, , . . . , ; in this case a ∈ [0 , . By also allowing rotations onthe disc D , we can take t = 0 in (3.3).) Using the complex coordinate x + i y in the plane df ( R ) = R × { } n − , the map (3.3) can be written as f ( z ) = (cid:18) c e i t z + a ae i t z , p, , . . . , (cid:19) = ( h ( z ) , p, , . . . , . chwarz–Pick lemma for harmonic maps which are conformal at a point 9From (3.2) it follows that | h ′ (0) | = c (1 − | a | ) = c − c | a | c = 1 − | q | sin θ − | q | cos θc = 1 − | q | p − | q | sin θ . Since f (0) = q , k df k = | h ′ (0) | , and θ is the angle between f (0) and df ( R ) , we haveequality in (2.2) at z = 0 .Lemma 4.1, proved in the following section, shows that every harmonic disc g : D → B n with the same centre g (0) = q = f (0) and direction dg ( R ) = Λ , with g conformal andorientation preserving at , satisfies | dg | ≤ | df | , with equality if and only if g is one of thediscs (3.3). This will prove Theorem 2.1.The proof of Lemma 4.1 uses ideas from Lempert’s seminal paper [19] concerningholomorphic discs F : D → Ω in a bounded strongly convex domain Ω ⊂ C n whichare extremal for the Kobayashi metric K Ω at the point F (0) ∈ Ω , in the sense that | F ′ (0) | is maximal among all holomorphic discs G : D → Ω with G (0) = F (0) and G ′ (0) = λF ′ (0) for some λ ∈ C . Lempert showed that every such disc F is properlyembedded in Ω , of Hölder class C / ( D ) , and is a complex geodesic of K Ω in the sensethat the size of its complex derivative F ′ ( z ) is maximal at every point z ∈ D (which meansthat K Ω ( F ′ ( z )) = 1 for all z ∈ D ); furthermore, the Kobayashi distance in Ω between anypair of points F ( a ) and F ( b ) equals the Poincaré distance between the points a and b in D .(Recall that the Kobayashi metric on D coincides with the Poincaré metric; see [15]).We now explain the connection with Theorem 2.1. The domain of interest is the tube(3.4) Ω = B n × i R n = { z = x + i y : x , y ∈ R n , | x | < } ⊂ C n . This domain is convex and strongly pseudoconvex, but is not bounded or strongly convex.Nevertheless, it is Kobayashi hyperbolic (since it is biholomorphic to a bounded domain),and Lempert’s methods apply with minor modifications; see the monograph by Jarnicki andPflug [12, Sect. 11.1] and the papers by Zaj ˛ac [24, 25]. A harmonic map f : D → B n ( n ≥ is the real part of a holomorphic map F : D → Ω , which is unique up to thechoice of ℑ F (0) . An elementary calculation using the Cauchy–Riemann equations and(2.1) shows that f is conformal at a point z ∈ D if and only if the complex derivative F ′ ( z ) satisfies the nullity condition P nk =1 F ′ k ( z ) = 0 (see Osserman [20, p. 30] or Duren[6, Chap. 9]). In particular, f is a conformal harmonic immersion if and only if F is animmersed holomorphic null disc , meaning that this nullity condition holds identically. (For n ≥ this gives the well-known correspondence between conformal minimal surfaces in R n and holomorphic null curves in C n .) Furthermore, assuming that f is conformal at ,we have that k df k ≥ k dg k for all harmonic discs g : D → B n such that g (0) = f (0) , dg ( R ) = df ( R ) , and g is conformal at , if and only if | F ′ (0) | ≥ | G ′ (0) | for allholomorphic discs G : D → Ω with G (0) = F (0) and G ′ (0) = λF ′ (0) for some λ ∈ C .Lemma 4.1 shows that this holds true if f is any one of the discs D → B n in Theorem 2.1;those lift to holomorphic null discs in Ω which are complex geodesics for the Kobayashimetric on Ω . They happen to be flat null discs. Complex geodesic in Ω which are not nulldiscs project to non-conformal harmonic maps D → B n .The correspondence between extremal conformal minimal discs in the ball B n ⊂ R n and Kobayashi geodesics in the tube B n × i R n ⊂ C n , described above, can be used in anybounded strongly convex domain D ⊂ R n with smooth boundary satisfying the following0 F. Forstneriˇc and D. Kalaj Condition N:
Given a point p ∈ D and a null vector = ν = ( ν , . . . , ν n ) ∈ C n (i.e., P ni =1 ν i = 0 ), the Kobayashi extremal disc in the tube T D = D × i R n through the point p + i ∈ T D in the direction ν is a holomorphic null disc.Indeed, our analysis, along with the proof of Lemma 4.1, implies the following. Theorem 3.1. If D is a bounded strongly convex domains in R n ( n ≥ with smoothboundary satisfying Condition N, then the extremal conformal harmonic discs in D are thereal parts of Kobayashi extremal holomorphic null discs in the tube T D = D × i R n . Problem 3.2.
Which bounded strongly convex domains in R n satisfy Condition N?Complex geodesics of the Kobayashi metric in tubes over convex domains D ⊂ R n werestudied by Zaj ˛ac [24, 25]. A closer analysis of these works may shed light on this problem.That Condition N holds on the ball B n may simply be a lucky coincidence which makes ouranalysis work. At any rate, this seems a fertile ground for further investigations.
4. The main lemma
As explained in Section 3, Theorem 2.1 is a consequence of the following lemma.
Lemma 4.1.
Let f : D → B n ( n ≥ be the disc (3.3) with t = 0 for some p ∈ [0 , , c = p − p ∈ (0 , , and a ∈ D . (If n = 2 , we assume that f is given by (3.3) withoutthe remaining components.) If g : D → B n is a harmonic disc such that g (0) = f (0) , g is conformal at , and dg ( R ) = df ( R ) , then | dg | ≤ | df | , with equality if and only if g ( z ) = f ( e it z ) or g ( z ) = f ( e it ¯ z ) for some t ∈ R and all z ∈ D .Proof. For simplicity of notation we assume that n = 3 ; the proof for n = 3 is exactly thesame. For n = 2 , we delete the remaining components and take c = 1 .Denote by h· , ·i the complex bilinear form on C n given by h z, w i = n X i =1 z i w i for z, w ∈ C n . Note that on vectors in R n this is the Euclidean inner product.Let f : D → B be the disc (3.3). Precomposing f by a suitable rotation of D we mayassume that t = 0 . Consider the holomorphic disc F : D → Ω = B × i R given by(4.1) F ( z ) = (cid:18) c z + a az , − c i z + a az , p (cid:19) , z ∈ D . Then, f = ℜ F . Suppose that g : D → B is a harmonic map such that g (0) = f (0) and g is a conformal immersion at z = 0 with dg ( R ) = df ( R ) . Up to replacing the map g ( z ) by g ( e i t z ) or g ( e i t ¯ z ) for some t ∈ R , we may assume that(4.2) dg = rdf for some r > . We must prove that r ≤ , and that r = 1 if and only if g = f .Let G : D → Ω be the unique holomorphic map with ℜ G = g and G (0) = F (0) . Inview of the Cauchy–Riemann equations, condition (4.2) implies(4.3) G ′ (0) = rF ′ (0) , chwarz–Pick lemma for harmonic maps which are conformal at a point 11where the prime denotes the complex derivative. It follows that the map ( F ( z ) − G ( z )) /z with values in C n is holomorphic on D , and its value at z = 0 equals(4.4) lim z → F ( z ) − G ( z ) z = F ′ (0) − G ′ (0) = (1 − r ) F ′ (0) . Since g : D → B is a bounded harmonic map, it has a nontangential boundary value atalmost every point of the circle T = b D . Since the Hilbert transform is an isometry on theHilbert space L ( T ) , the same is true for the holomorphic map G (see Garnett [9]).Note that for each z = e i t ∈ b D the vector f ( z ) ∈ b B is the unit normal vector to thesphere b B at the point f ( z ) . Since B is strongly convex, we have that(4.5) ℜ (cid:10) F ( z ) − G ( z ) , f ( z ) (cid:11) = (cid:10) f ( z ) − g ( z ) , f ( z ) (cid:11) ≥ a.e. z ∈ b D , and the value is positive for almost every z ∈ b D if and only if g = f . It is at this point thatstrong convexity of the ball B is used in an essential way.Let us now consider the function ˜ f on the circle b D given by(4.6) ˜ f ( z ) = z | az | f ( z ) , | z | = 1 . Explicit calculation, taking into account z ¯ z = 1 , shows that(4.7) ˜ f ( z ) = c (cid:0) a + 4( ℜ a ) z + (1 + ¯ a ) z (cid:1) c (cid:0) i (1 − a ) + 4( ℑ a ) z + i (¯ a − z (cid:1) p ( z + a )(1 + ¯ az ) . We extend ˜ f to all z ∈ C by letting it equal the quadratic holomorphic polynomial map onthe right hand side above. Since | az | > for z ∈ D , (4.5) implies h ( z ) := ℜ (cid:10) F ( z ) − G ( z ) , | az | f ( z ) (cid:11) = (cid:10) f ( z ) − g ( z ) , | az | f ( z ) (cid:11) ≥ a.e. z ∈ b D , and h > almost everywhere on b D if and only if g = f . From (4.6) we see that(4.8) h ( z ) = ℜ (cid:28) F ( z ) − G ( z ) z , ˜ f ( z ) (cid:29) a.e. z ∈ b D Since the maps ( F ( z ) − G ( z )) /z and ˜ f ( z ) are holomorphic on D , the formula (4.8) providesan extension of h from b D to a nonnegative harmonic function on D which is positive on D unless f = g . Inserting the value (4.4) into (4.8) gives h (0) = ℜ (cid:10) F ′ (0) − G ′ (0) , ˜ f (0) (cid:11) = (1 − r ) ℜ (cid:10) F ′ (0) , ˜ f (0) (cid:11) ≥ , with equality if and only if f = g . Applying this argument to the linear map g ( z ) = f (0) + rdf ( z ) ( z ∈ D ) for a small r > we get ℜ h F ′ (0) , ˜ f (0) i > . It follows that r ≤ , with equality if and only if g = f . (cid:3) Remark 4.2 (Stationary discs) . The above proof shows that any holomorphic disc F in Ω = B × i R of the form (4.1) is a stationary disc in Ω . Indeed, in Lempert’s terminologyfrom [19], a proper holomorphic disc in a bounded strongly convex domain Ω ⊂ C n withsmooth boundary is stationary disc if ν ( z ) is the unit normal to b Ω along the boundary circle F ( b D ) , then there is a positive function q ( z ) > on b D such that the function zq ( z ) ν ( z ) extends from the circle | z | = 1 to a holomorphic function ˜ f ( z ) on the disc D . The use ofsuch a function, along with convexity of the domain, enables the arguments used above to2 F. Forstneriˇc and D. Kalajshow that every stationary disc F is the unique Kobayashi extremal disc through the point F ( a ) in the tangent direction F ′ ( a ) for every a ∈ D . In our case we have ν ( z ) = f ( z ) ,which is real-valued, and a suitable holomorphic function ˜ f is given by (4.6) and (4.7).The fact that Ω is an unbounded tube does not matter since the discs (4.1) lift to properholomorphic null discs in Ω without any boundary points at infinity. (cid:3) Remark 4.3 (The role of conformality) . A key hypotheses in Theorem 2.1 is that the givenharmonic map D → B n is conformal at the reference point, which we may assume tobe the origin ∈ D . Without this condition, our proof of Lemma 4.1 (and hence ofTheorem 2.1) fails. The reason is that conditions (4.2) and (4.3) no longer hold if f isthe extremal conformal disc (3.3) and g is a harmonic map satisfying g (0) = f (0) and dg ( R ) = df ( R ) , but g is not conformal at . Indeed, denoting by F and G holomorphicdiscs D → Ω = B n × i R n such that f = ℜ F and g = ℜ G , the vectors F ′ (0) and G ′ (0) in C n have different directions, and hence the proof of Lemma 4.1 fails. (cid:3)
5. Growth of conformal minimal discs
In this section we prove part (b) of Theorem 2.3. As mentioned earlier, part (a) is aconsequence of Theorem 2.1, and the estimates in (a) will be used to prove (b).Assume that f : D → B n for n ≥ is a conformal minimal disc with f (0) = 0 . Fixa point z ∈ D \ { } ; we wish to prove that | f ( z ) | ≤ | z | . By a rotation on D we mayassume that z = r ∈ (0 , . Set ρ ( x ) = | f ( x ) | for x ∈ I := ( − , ; so we must provethat ρ ( r ) ≤ r . We shall denote the derivative on the variable x by a prime. We have that ρ ( x ) ρ ′ ( x ) = ddx ρ ( x ) = ddx f ( x ) · f ( x ) = 2 f ( x ) · f ′ ( x ) = 2 | f ( x ) | · | f ′ ( x ) | cos φ ( x ) , where φ ( x ) ∈ [0 , π ] is the angle between the vectors f ( x ) and f ′ ( x ) . Let θ ( x ) ∈ [0 , π/ denote the angle between f ( x ) and the -plane Λ x = df x ( R ) . Since f ′ ( x ) ∈ Λ x , we havethat φ ( x ) ≥ θ ( x ) , with equality if and only if the orthogonal projection of f ( x ) to Λ x isparallel to f ′ ( x ) and points in the same direction. Therefore, cos φ ( x ) ≤ cos θ ( x ) holds forall x ∈ I . Note also that k df x k = | f ′ ( x ) | since f is conformal. By using this informationin the estimate (2.2) for k df x k , the above identity implies ρ ( x ) ρ ′ ( x ) ≤ | f ( x ) | − | f ( x ) | − x · cos θ ( x ) p − | f ( x ) | sin θ ( x ) ≤ ρ ( x ) 1 − ρ ( x ) − x · cos θ ( x ) p − ρ ( x ) sin θ ( x ) . Since ρ ( x ) < , we have that cos θ ( x ) p − ρ ( x ) sin θ ( x ) ≤ , with equality if and only if ρ ( x ) = 0 , or ρ ( x ) > and θ ( x ) = 0 . Hence, the previousinequality gives ρ ′ ( x ) ≤ − ρ ( x ) − x , or equivalently dρ ( x )1 − ρ ( x ) ≤ dx − x . Integrating from x = 0 to x = r and taking into account that ρ (0) = 0 gives ρ ( r ) = | f ( r ) | ≤ r . In view of our normalization, this proves that | f ( z ) | ≤ | z | for all z ∈ D .chwarz–Pick lemma for harmonic maps which are conformal at a point 13The proof also shows that | f ( r ) | = r holds if and only if θ ( x ) = φ ( x ) = 0 at all points x ∈ [0 , r ] where f ( x ) = 0 . If f ( x ) = 0 on a nontrivial subinterval, the identity principleimplies f ( x ) = 0 for all x ∈ I , and hence equality fails. If on the other hand f doesnot vanish on any subinterval of [0 , r ] , then θ ( x ) = φ ( x ) = 0 for all x . This means that f ′ ( x ) is parallel to f ( x ) for all x , and hence the image of the interval [0 , is a straight linesegment in B n . After an orthogonal rotation on R n we may assume that this segment liesin [0 , × R n − . Furthermore, the inequalities in the proof are then equalities, and hence f ( x ) = ( x, , . . . , for all x ∈ I . It follows that | df | = | f x (0) | = 1 , and hence (2.2) isan equality (and θ = 0 ). The last part of Theorem 2.1 then implies that f is a conformalparameterization of the linear disc df ( R ) ∩ B n . This completes the proof of Theorem 2.3. Remark 5.1.
From Corollary 2.6 and Theorem 2.8 we obtain precise distance estimates foran arbitrary conformal harmonic map f : M → B n from a hyperbolic Riemann surface M .The argument is similar to the one that we gave above in the special case. The conclusionis that f maps the Poincaré ball in M of radius r > centred at a given point p ∈ M intothe ball of the same radius r in B n centred at f ( p ) in the minimal metric (2.6). Explicitformulas for these balls are given by (2.11) in the disc D and (2.12) in the ball B n . (cid:3)
6. Schwarz lemma for not necessarily conformal harmonic discs
In this section we prove Theorem 2.2 and give some related examples.Precomposing the harmonic map f : D → B n in Theorem 2.2 by a suitable holomorphicautomorphism of D , we see that it suffices to prove the estimate (2.3) for z = 0 .Assume first that f : D → R is a harmonic function on the disc D = {| z | < } . Let F ( z ) be the holomorphic function on D with ℜ F = f and F (0) = f (0) ∈ R . F ( z ) = a + a z + a z + . . . be its Taylor series expansion. Writing z = re i t with ≤ r < and t ∈ R , we have f ( re i t ) = 14 (cid:0) a + a re i t + r e i t + · · · + a + ¯ a re − i t + ¯ a r e − i t + · · · (cid:1) = a + 12 ∞ X k =1 r k | a k | + · · · , where each of the remaining terms in the series contains a power e m i t for some m ∈ Z \{ } .Integrating around the circle | z | = r annihilates all such terms and yields Z π f ( re i t ) dt π = a + 12 ∞ X k =1 r k | a k | . Clearly, a = f (0) . Writing z = x + i y , we have that a = F ′ (0) = F x (0) = f x (0) − i f y (0) by the Cauchy–Riemann equations. Therefore, a = f (0) , | a | = f x (0) + f y (0) = |∇ f (0) | , and hence(6.1) Z π f ( re i t ) dt π = | f (0) | + 12 |∇ f (0) | r + 12 ∞ X k =2 r k | a k | . f = ( f , . . . , f n ) : D → B n is a harmonic map. Then, P nj =1 f j ( re i t ) < for all ≤ r < and t ∈ R . Integrating this inequality and taking intoaccount the identity (6.1) for each component f j of f gives Z π | f ( re i t ) | dt π = | f (0) | + 12 |∇ f (0) | r + 12 ∞ X k =2 r k | a k | < . Letting r increase to gives | f (0) | + |∇ f (0) | ≤ , with equality if and only if all higherorder coefficients in the Fourier expansion of f vanish. The latter holds if and only if f is alinear disc. This gives the estimate (2.3).Note that (2.3) holds if the L -Hardy norm of f is at most . This does not necessarilyimply that there is a harmonic disc in B n reaching equality in (2.3). However, equalityis reached if f (0) is orthogonal to the -plane df ( R ) . In this case we may assume that f (0) = (0 , , p, . . . , for some ≤ p < and df ( R ) = R × { } n − . The affine disc Σ = (cid:8) ( x, y, p, , . . . ,
0) : x + y < − p (cid:9) of radius c = p − p is then orthogonal to f (0) , proper in B n , and its conformal linearparameterization f has gradient of size c √ at the origin, so | f (0) | + |∇ f (0) | = p + c = 1 . (Compare with (3.1) and (3.3).) This completes the proof of Theorem 2.2.We now show by examples that the inequality (2.2) fails in general for somenonconformal harmonic maps and even harmonic diffeomorphisms of the disc. Example 6.1.
Let U be the harmonic function on the disc D given by U ( z ) = ℑ π log 1 + z − z = 2 π arctan 2 y − x − y . This is the extremal harmonic function whose boundary value equals +1 on the upper unitsemicircle and − on the lower semicircle, and we have that ∇ U (0) = 4 π (0 , , |∇ U (0) | = 4 π . For every c ∈ R the harmonic map f ( z ) = 1 p | c | (cid:0) c + i U ( z ) (cid:1) , z ∈ D clearly takes the unit disc into itself. For c = 1 we have f (0) = √ , ∇ f (0) = √ π (cid:18) (cid:19) , |∇ f (0) | = 2 √ π ≈ . , √ (cid:0) − | f (0) | (cid:1) = √ ≈ . . Hence, the inequality (2.2) fails in this example. On the other hand, √ p − | f (0) | = 1 ,so the inequality (2.3) holds, as it should by Theorem 2.2.With some more effort we can show that the inequality (2.2) fails for harmonicdiffeomorphisms of the unit disc onto itself. Consider the sequence ϕ n ( n ∈ N ) of sense-preserving homeomorphisms of the interval [0 , π ] onto itself, defined by ϕ n ( t ) = (cid:26) π π − /n t, if t ∈ [0 , π − /n ] ; (cid:0) π − nπ (cid:1) + nπt, if t ∈ [2 π − /n, π ] .chwarz–Pick lemma for harmonic maps which are conformal at a point 15Let φ n : T → T be the associated sequence of homeomorphisms of the circle T = b D givenby φ n ( e i t ) = e i ϕ n ( t ) for t ∈ [0 , π ] . Denote by f n ( z ) = P [ φ n ]( z ) = 12 π Z π − | z | | e i t − z | φ n ( e i t ) dt, z ∈ D the Poisson extension of φ n . By Radó–Kneser–Choquet theorem (see [6, Sect. 3.1]), f n is a harmonic diffemorphism of D for every n ∈ N . As n → ∞ , the sequence f n converges uniformly on compacts in D to the harmonic map f = P [ φ ]( z ) , where φ ( e i t ) = lim n →∞ φ n ( e i t ) = e i t/ for t ∈ [0 , π ) . Further, lim n →∞ |∇ f n (0) | − | f n (0) | = |∇ f (0) | − | f (0) | . A calculation shows that √ |∇ f (0) | − | f (0) | = p | A | + | B | − | C | , where A = 1 π Z π e i t cos t dt = − i π , B = 1 π Z π e i t sin t dt = 83 π , and C = 12 π Z π e i t/ dt = 2 i π . Hence, √ |∇ f (0) | − | f (0) | = 2 √ π (cid:0) − π (cid:1) ≈ . . This show that (2.2) fails for harmonic diffeomorphisms of the unit disc onto itself. (cid:3)
7. A Schwarz–Pick lemma for quasiconformal harmonic maps
In this section we apply the Schwarz–Pick lemma for harmonic self-maps of the disc,given by Theorem 1.1, to provide an estimate of the differential of a harmonic map f : D → D in terms of its second Beltrami coefficient (7.1) ω ( z ) = f ¯ z f z , z ∈ D . Here, f z = ( f x − i f y ) and f ¯ z = ( f x + i f y ) . We begin with a brief motivation, referringto Duren [6] for background on this topic.If the map f is harmonic then ω is a holomorphic function (see (7.2)). This is not thecase for the Beltrami coefficient µ from the Beltrami equation f ¯ z = µ ( z ) f z . The number | µ ( z ) | = | ω ( z ) | measures the dilatation of df z ; in particular, µ ( z ) = ω ( z ) = 0 if andonly if f is conformal at z . A map f : D → C is said to be quasiconformal if theessential supremum of the dilatation is smaller than one: k µ k ∞ = k ω k ∞ < . Sincethe Jacobian of f equals J f = | f z | − | f ¯ z | , this implies that f preserves the orientation atevery point where f z = 0 ; such maps are called sense preserving . A fundamental result ofquasiconformal theory says that for every measurable function µ on D with k µ k ∞ < thereexists a quasiconformal homeomorphism f : D → Ω onto a given simply connected domain Ω ( C satisfying the Beltrami equation f ¯ z = µf z . In the same spirit, Hengartner andSchober [11] (see also [6, Chapter 7]) studied the existence of harmonic maps f : D → Ω with a given holomorphic second Beltrami coefficient ω = ω f : D → D . Such maps exist6 F. Forstneriˇc and D. Kalajand are quasiconformal provided that k ω k ∞ < , and they map the unit disc onto Ω . If k ω k ∞ = 1 , they exist but fail to be quasiconformal, and may fail to be surjective.The following result generalizes the Schwarz–Pick lemma given by Theorem 1.1. Theorem 7.1.
Assume that f is a sense preserving harmonic map of the unit disc into itself,and let ω ( z ) denote its second Beltrami coefficient (7.1) . Then we have the inequality √ |∇ f ( z ) | ≤ ℜ ( ω ( z ) f ( z ) )(1 − | ω ( z ) | )(1 − | z | ) + 1 + | ω ( z ) | − | ω ( z ) | − | f ( z ) | − | z | , z ∈ D . If f is conformal at a point z , i.e. ω ( z ) = 0 , this coincides with the Schwarz–Pickinequality (1.1) in Theorem 1.1. The estimate is not sharp in general. For example, if f (0) = 0 then the left hand side is at most by Theorem 2.2, but the right hand side isat least and equals only if f is conformal at . Hence, the inequality is trivial in thiscase. However, it is nontrivial at any point where f ( z ) = 0 . In this connection, we mentionthe recent work of Kovalev and Yang [16] and Brevig et al. [4] giving conditions on a pairof complex numbers α, β ∈ C for which there exists a harmonic map f : D → D with f (0) = 0 satisfying f z (0) = α and f ¯ z = β . We do not see a direct relationship of theirwork to Theorem 7.1, which is of interest only in the case when f (0) = 0 . Proof.
It suffices to prove the inequality in the theorem for z = 0 . For other points, weobtain it replacing f by f ◦ ϕ z for ϕ z ∈ Aut( D ) . However, we cannot reduce to the case f (0) = 0 since postcompositions by automorphisms of D are not allowed. The main idea isto construct from f a new harmonic map ˜ f : D → D which is conformal at , to which wethen apply the Schwarz–Pick lemma given by Theorem 1.1.Let us write f = g + h where g and h are holomorphic functions on D . Then,(7.2) f z ( z ) = g ′ ( z ) , f ¯ z ( z ) = h ′ ( z ) , ω ( z ) = h ′ ( z ) /g ′ ( z ) . We see in particular that the second Beltrami coefficient ω is holomorphic. It follows that(7.3) |∇ f | = | f x | + | f y | = 2 (cid:0) | f z | + | f ¯ z | (cid:1) = 2 (cid:0) | g ′ | + | h ′ | (cid:1) = 2 | g ′ | (1 + | ω | ) . Since f is sense preserving, we have that | g ′ ( z ) | ≥ | h ′ ( z ) | . Let a = g ′ (0) and b = h ′ (0) .The Cauchy–Schwarz inequality shows that the harmonic function ˜ f ( z ) = ¯ af − ¯ b ¯ f p | a | + | b | , z ∈ D maps the unit disc into itself. We have ˜ f = ˜ g + ˜ h , where ˜ g = ¯ ag − ¯ bh p | a | + | b | and ˜ h = ah − bg p | a | + | b | are holomorphic functions. Since(7.4) ˜ h ′ (0) = ah ′ (0) − bg ′ (0) p | a | + | b | = 0 , ˜ f is conformal at z = 0 . From our Schwarz–Pick lemma (see Theorem 1.1) we get(7.5) √ − |∇ ˜ f (0) | ≤ − | ˜ f (0) | . chwarz–Pick lemma for harmonic maps which are conformal at a point 17Taking into account (7.3) and (7.4) we have that √ − |∇ ˜ f (0) | = q | ˜ f z (0) | + | ˜ f ¯ z (0) | = q | ˜ g ′ (0) | + | ˜ h ′ (0) | = | ˜ g ′ (0) | = | a | − | b | p | a | + | b | = | g ′ (0) | − | h ′ (0) | p | g ′ (0) | + | h ′ (0) | . Together with (7.5) this gives the estimate | g ′ (0) | − | h ′ (0) | p | g ′ (0) | + | h ′ (0) | ≤ − | g ′ (0) f (0) − h ′ (0) f (0) | | g ′ (0) | + | h ′ (0) | ≤ ℜ (cid:16) g ′ (0) h ′ (0) f (0) (cid:17) | g ′ (0) | + | h ′ (0) | + 1 − | f (0) | . In view of (7.2), this inequality can be written in the form − | ω (0) | p | ω (0) | | g ′ (0) | ≤ ℜ ( ω (0) f (0) )1 + | ω (0) | + 1 − | f (0) | . From (7.3) we see that | g ′ (0) | = |∇ f (0) |√ p | ω (0) | . Inserting this into the expression on the left hand side of the previous inequality gives |∇ f (0) |√ − | ω (0) | | ω (0) | ≤ ℜ ( ω (0) f (0) )1 + | ω (0) | + 1 − | f (0) | . which is clearly equivalent to √ |∇ f (0) | ≤ ℜ ( ω (0) f (0) )1 − | ω (0) | + 1 + | ω (0) | − | ω (0) | (cid:0) − | f (0) | (cid:1) . This completes the proof. (cid:3)
Acknowledgements.
F. Forstneriˇc is partially supported by research program P1-0291and grant J1-9104 from ARRS, Republic of Slovenia. D. Kalaj is partially supportedby a research fund of University of Montenegro. We wish to thank Antonio Alarcónfor his remarks which led to improved presentation, Dmitry Khavinson for informationregarding harmonic self-maps of the disc, Joaquín Pérez for consultation on the state ofthe art concerning growth of conformal minimal discs in the ball, and László Lempertand Sylwester Z ˛ajac for communications concerning Kobayashi geodesics in convex tubedomains. The first named author also thanks Miodrag Mateljevi´c with whom he discussedthe problem of estimating derivatives of conformal minimal discs in 2016.
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Franc ForstneriˇcUniversity of Ljubljana, Faculty of Mathematics and Physics, Jadranska 19, SI–1000Ljubljana, Slovenia, and Institute of Mathematics, Physics and Mechanics, Jadranska 19,SI–1000 Ljubljana, Sloveniae-mail: [email protected]