1 Introduction and statement of the main result

Transcription

1 INNER ESTIMATE AND QUASICONFORMAL HARMONIC MAPS BETWEEN SMOOTH DOMAINS By DAVID KALAJ AND MIODRAG MATELJEVIĆ Abstract. We prove a type of inner estimate for quasi-conformal diffeomorphisms, which satises a certain estimate concerning their Laplacian. This, in turn, implies that quasiconformal harmonic mappings between smooth domains (with respect to an approximately analytic metric), have bounded partial derivatives; in particular, these mappings are Lipschitz. We discuss harmonic mappings with respect to (a) spherical and Euclidean metrics (which are approximately analytic) (b) the metric induced by a holomorphic quadratic differential. 1 Introduction and statement of the main result 1.1 Basic facts and notation. Let U and H denote the unit disc and the upper half plane, respectively. By Ω and D we denote simply connected domains. Suppose that fl is a rectiable curve in the complex plane or on the Riemann sphere S 2. Denote by l the length of fl, and let :[0;l]! fl be the natural parameterization of fl, i.e., the parameterization satisfying the condition j _ (s)j =1 for almost all s 2 [0;l]: We say that fl is of class C n;μ, for n 2 N, 0 <μ» 1 if is of class C n and sup t;s j (n) (t) (n) (s)j jt sj μ < 1: We call Jordan domains in C bounded by C n;μ Jordan curves C n;μ domains or smooth domains. Let ρ(w )jdwj 2 be an arbitrary conformal C 1 -metric dened on D. Iff :Ω! D is a C 2 mapping between the Jordan domains Ω and D, the energy integral of f is dened by the formula (1.1) E[f; ρ] = Z Ω ρ f f (jf z j 2 + jf μz j 2 )dx dy: JOURNAL D ANALYSE MATHÉMATIQUE, Vol. 99 (2006) 117

2 118 D. KALAJ AND M. MATELJEVIĆ The stationary points of the energy integral satisfy the Euler Lagrange equation (1.2) f zz +(logρ) w f ff z fμz =0; and a C 2 solution of this equation is called a harmonic mapping (more precisely, a ρ-harmonic mapping). It is known that f is a harmonic mapping if and only if the mapping (1.3) Φ=ρ f ff z f μz is analytic. If ' is a holomorphic mapping different from 0 and if ρ = j'j on D,we call ρ a '-metric. We will call the corresponding harmonic mapping '-harmonic. Notice that for ρ =1,aρ-harmonic mapping is a Euclidean harmonic function. Let 0» k < 1 and K = 1+k. An orientation preserving diffeomorphism 1 k f :Ω! D between two domains Ω;D ρ C is called a K or a k-quasiconformal mapping (briefly, a q.c. mapping) if it satises the condition (1.4) jf z (z)j»kjf z (z)j for each z 2 Ω: In this paper, we mainly consider harmonic quasiconformal mappings between smooth domains. 1.2 Background. The rst characterization of harmonic quasiconformal mappings with respect to the Euclidean metric for the unit disc was given by O. Martio [17]. Below are several important results in this area. Theorem P ([22]). If w is a harmonic diffeomorphism of the unit disc onto itself, then the following conditions are equivalent: w is q.c.; w is bi-lipschitz; the boundary function is bi-lipschitz and the Hilbert transformation of its derivative is in L 1. Theorem KP ([13] and [9]). An orientation-preserving homeomorphism ψ of the real axis can be extended to a q.c. harmonic homeomorphism of the upper half-plane if and only if ψ is bi-lipschitz and the Hilbert transformation of the derivative ψ 0 is bounded. Theorem K ([12]). If Ω and Ω 0 are Jordan domains with C 1;μ boundary (0 < μ» 1), then every quasiconformal harmonic function from Ω onto Ω 0 is Lipschitz. If in addition Ω 0 is convex, then w is bi-lipschitz. Moreover if w : Ω! Ω 0 is a harmonic diffeomorphism, where Ω is the unit disc and Ω 0 is a convex domain with C 1;μ boundary, then the following conditions are equivalent: w is quasiconformal; w is bi-lipschitz; the boundary function is bi-lipschitz and

3 INNER ESTIMATE AND Q.C. HARMONIC MAPS BETWEEN SMOOTH DOMAINS 119 the Hilbert transformation of its derivative is in L 1 ; and therefore a harmonic diffeomorphism in this setting is a quasi-isometry with respect to the corresponding Poincaré distance. Concerning quasi-isometry the second author in joint work with M. Knežević obtained the right constant. They proved Theorem MK1 ([15]). If f is a K q.c. harmonic diffeomorphism from the upper half plane H onto itself and f (1) =1, then jz 1 z 2 j=k»jf (z 1 ) f (z 2 )j»kjz 2 z 1 j; where z 1 ;z 2 2 H and f is a (1=K; K) quasi-isometry with respect to the Poincaré distance. Theorem MK2 ([15]). If f is a K q.c. harmonic diffeomorphism from the unit disc U onto itself, then f is a (1=K; K) quasi-isometry with respect to the Poincaré distance: d h (z 1 ;z 2 )=K» d h (f (z 1 ); f (z 2 ))» Kd h (z 1 ;z 2 ); where d h is hyperbolic distance in the unit disc. Concerning hyperbolic q.c. harmonic mappings, we present here two results. Theorem W1 ([28]). Every harmonic quasi-conformal mapping from the unit disc onto itself is a quasi-isometry of the Poincaré disc. Theorem W2 ([28]). A harmonic diffeomorphism of the hyperbolic plane H 2 is quasiconformal if and only if its Hopf differential (i.e., the function Φ dened in (1.3)) is uniformly bounded with respect to the Poincaré metric. For the other results in this area, see [19], [15], [16], [27], [24], [25], [21] and [2]. 1.3 New results. The following proposition plays an important role in [9], [13] and [15]. Proposition 1.1. Let f be an Euclidean harmonic 1 1 mapping of the upper half-plane H onto itself, continuous on H, normalized by f (1) = 1, and let v = Imf. Then v(z) = cy, where c is a positive constant. In particular, v has bounded partial derivatives on H. Suppose that f is a harmonic Euclidean mapping of the unit disc onto a smooth domain D and ψ is a conformal mapping of D onto H. The composition ψ f f

4 120 D. KALAJ AND M. MATELJEVIĆ is rarely Euclidean harmonic, so we cannot apply Proposition 1.1. However, the composition is harmonic with respect to the metric ρ(w) =j(ψ 1 ) 0 (w)j 2 (see Corollary 4.3). Having this in mind, our idea is to apply the following result instead of Proposition 1.1 in more complicated cases. Proposition 1.2 (Inner Estimate). (Heinz Bernstein; see [8]). Let s : U! R be a continuous function from the closed unit disc U into the real line satisfying the conditions (1) s is C 2 on U, (2) s b ( ) =s(e i ) is C 2, and (3) j sj»c 0 j5sj 2 on U for some constant c 0. Then the function j5sj = jgrad sj is bounded on U. We refer to this result as the inner estimate. Applying this estimate and results of Kellogg Warschawski (see below), we prove the main result of this paper. Theorem 1.3. Let f be a quasiconformal C 2 diffeomorphism from the C 1;ff Jordan domain Ω onto the C 2;ff Jordan domain D. If there exists a constant M such that (1.5) j fj»mjf z f μz j ; z 2 Ω; then f has bounded partial derivatives. In particular, it is a Lipschitz mapping. In particular, Theorem 1.3 holds if h is quasiconformal ρ-harmonic and the metric ρ is approximately analytic, i.e., j on Ω (see Theorems 3.1, 3.3, 3.5 below). Since Euclidean and spherical metrics are approximately analytic, our results can be viewed as extensions of the results in [17], [22], [13], [9] and [12] (mentioned in Subsection 1.2). The main result is proved in Section 2, and its applications are given in Section 3. In Section 4, we show that the composition of a conformal mapping ψ and a '-harmonic mapping satises certain properties (see Theorem 4.1). In particular, if f is a natural parameter, we obtain a representation of '-harmonic mappings by means of Euclidean harmonics. We also provide some examples of '-harmonic mappings and prove that Theorem 3.1 holds for more general domains. 2 The proof of the main result To prove the main result we need the following two results. Proposition 2.1 (Kellogg [7]). If the domain D = Int( ) is C 1;ff and! is a conformal mapping of U onto D, then! 0 and log! 0 are in Lip ff. In particular, j! 0 j is bounded from above and below on U.

5 INNER ESTIMATE AND Q.C. HARMONIC MAPS BETWEEN SMOOTH DOMAINS 121 Proposition 2.2 (Kellogg and Warschawski [23, Theorem 3.6]).. If the domain D = Int( ) is C 2;ff and! is a conformal mapping of U onto D, then j! 00 j has a continuous extension to the boundary. In particular, it is bounded from above on U. Proof of Theorem 1.3. Let g be a conformal mapping of the unit disc onto Ω. Let f ~ = f f g. Since ~f = jg 0 j 2 f and f ~ z ~f μz = jg 0 j 2 f z f μz, f ~ satises the inequality (1.5). We prove the theorem for f ~ and then apply Kellogg s theorem. For simplicity, we write f instead of f. ~ Let t = T be an arbitrary xed point. Step 1 (Local Construction). In this step, we show that there are two Jordan domains D 1 and D 2 in D with C 2;ff boundary such that (i) D 1 ρ D 2 ρ 2 is a connected arc containing the point w = f (t) in its interior, (iii) ; 2 1 ρ D. Let H 1 be the Jordan domain bounded by the Jordan curve fl 1 which is composed by the following sequence of Jordan arcs: fy 1=5 +(2 x) 1=5 =1; 1» x» 2g; f(2 y) 1=5 +(2 x) 1=5 =1; 1» x» 2g; [(1; 2); ( 1; 2)]; f(2 y) 1=5 +(2+x) 1=5 =1; 2» x» 1g; fy 1=5 +(2+x) 1=5 =1; 2» x» 1g and [( 1; 0); (1; 0)]: Let H 2 be the Jordan domain bounded by the Jordan curve fl 2 which is composed by the following sequence of Jordan arcs: fy 1=5 +(2 x) 1=5 =1; 1» x» 2g; [(2; 1); (2; 2)]; f(3 y) 1=5 +(2 x) 1=5 =1; 1» x» 2g; [(1; 3); ( 1; 3)]; f(3 y) 1=5 +(2+x) 1=5 =1; 2» x» 1g; [( 2; 2); ( 2; 1)]; fy 1=5 +(2+x) 1=5 =1; 2» x» 1g and [( 1; 0); (1; 0)]: Note that H 1 ρ H 2 ρ [ 2; 2] [0; 1 R 2 R = [ 1; 1] and 1 ;@H 2 2 C 3. Let be an orientation preserving arc-length parameterization of fl such that for s 0 2 (0; length(fl)), (s 0 ) = f (t). Let D Λ = 0 (s 0 )D, b = 0 (s 0 )f (t) and Λ = 0 (s 0 ). Then there exists r > 0 such that (b; b + ir] ρ D Λ. Since fl Λ Λ 2 C 2;ff, it follows that, there exist x 0 > 0, ">0, y 0 2 (0;r=3), ac 2;ff function h :[ 2x 0 ; 2x 0 ]! R, h(0) = 0, and the domain D2 Λ ρ D Λ such that (1) Λ ([s 0 "; s 0 + "]) = fb +(x; h(x)) : x 2 [ 2x 0 ; 2x 0 ]g, (2) D2 Λ = fb +(x; h(x) +y) : x 2 [ 2x 0; 2x 0 ];y2(0; 3y 0 ]g.

6 122 D. KALAJ AND M. MATELJEVIĆ Let :[ 2; 2] [0; 3]! D Λ 2 be the mapping dened by (x; y) =b +(xx 0 ;h(xx 0 )+yy 0 ): Then is a C 2;ff diffeomorphism. Take D i = 0 (s 0 ) (H i ), i = 1; 2. Obviously, D 1 ρ D 2 ρ D and D 1 and D 2 have C 2;ff boundary. Observe that f (t) = 0 (s 0 ) 0 (s 0 )f (t) = 0 (s 0 ) (0) 2 0 (s 0 ) ([ 1; 1]) 2 : Step 2 (Application of the Inner Estimate). Let f be a conformal mapping of D 2 onto H such that f 1 (1) 2 1. Let Ω 1 = f(d 1 ). Then there exist real numbers a; b; c; d such that a<c<d<b, [a; b] =@Ω 1 R and l = f 1 (@Ω 1 n [c; d]) ρ D: Let U 1 = f 1 (D 1 ) and be a conformal mapping between the unit disc and the domain U 1. Then the mapping ^f = f f f f is a C 2 diffeomorphism of the unit disc onto the domain Ω 1 such that (a) ^f is continuous on the boundary T (it is q.c.) and (b) ^f is C 2 on the set T 1 = ^f 1 (@Ω 1 n (c; d)). Let s := Im ^f. First, note that (a) implies that s is continuous on T On the other hand, as ^f 2 C 2, (1) s 2 C 2 (U). From (b), we obtain that s is C 2 on the set T 1 = ^f 1 (@Ω n (c; d)). Furthermore, s =0on T 2 = ^f 1 (a; b); and therefore s is C 2 on T 2 = ^f 1 (a; b). Hence (2) s is C 2 on T = T 1 [ T 2. In other words, the function s b : R! R dened by s b ( ) =s(e i ) is C 2 in R. In order to apply the inner estimate, we have to prove that (3) j s(z)j»c 0 j5s(z)j 2, z 2 U; where c 0 is a constant. Now (2.1) s z = ^f z ^f μz 2i and s μz = ^f μz ^f z ; 2i so s z = s μz ; therefore (2.2) js z j 2 = js μz j 2 = j5sj2 : 2

7 INNER ESTIMATE AND Q.C. HARMONIC MAPS BETWEEN SMOOTH DOMAINS 123 Since ^f = f f f f, we obtain ^f zμz ^f z ^fμz f 00 = f f μ@f : As ^f is a k-q.c. mapping, we have j sj = jim ^fj f»j^f z j j^fμzj 00 f f μ@f» kj ^f z j 2 f 00 f f μ@f : Using (2.1) and (2.2) respectively, we obtain and (2.3) j sj» 2k (1 k) 2 (1 k)j ^f z j»2js z j; f 00 f f μ@f j5sj2 : Proposition 2.1 and Proposition 2.2 imply that the function jf 0 j is bounded from below by a positive constant C1 and the function jf 00 j is bounded from above by a constant C2. With the help of (1.5), we obtain (2.4) Combining (2.3) and (2.4), we have f 00 f f μ@f» 4C 2 + MC1 4C1 2 : j sj»c0j 5sj 2 ; where 2k c0 = (1 k) 2 4C 2 + MC1 4C1 2 : Proposition 1.2 implies that the function j5sj is bounded by a constant b t. Since ^f is a k q.c. mapping, we have Finally, (1 k)j ^f z j»j^f z ^fμzj»2js z j» p 2b t : j ^f z j + j ^fμzj» p 2 1+k 1 k b t: Let T t = (f f ) 1 (c; d). Observe that c and d depend on the xed point t. c;d Since t 2 T t, we obtain T = S c;d t2t T t ; and therefore there exists a nite set c;d ft1;:::;t n g such that T = S n i=1 T ti. c;d Since the mapping i = is conformal and maps the circular arc T i = (fff f ) 1 (a; b) onto the circular arc (fff ) 1 (a; b), it can be conformally extended

8 124 D. KALAJ AND M. MATELJEVIĆ across the arc T 0 =(fff f ) 1 [c; d]. Hence, there exists a constant A i i such that j 0 (z)j 2A i on Ti. 0 It follows that there exists r i 2 (0; 1) such that j 0 (z)j A i in T i = fρz : z 2 T 0;r i i» ρ» 1g. By Proposition 2.1, the conformal mapping f i = f and its inverse have C 1 extensions to the boundary. Therefore, there exists a positive constant B i such that jf 0 (z)j B i on some neighborhood of f 1 [c; d] with respect to D. Thus, the mapping f = f 1 f ^f f 1 has bounded derivative in some neighborhood of the set T ti, on which it is bounded by the constant c;d Set C 0 =maxfc 1 ;:::;C n g. Then C i = p2 1+k b ti : 1 k A i B i jf z (z)j + jf μz(z)j»c 0 for all z 2 U near T As f is diffeomorphism in U, we obtain the desired conclusion. Λ 3 Applications Let D be a domain in C and ρ a conformal metric in D. The Gaussian curvature of the domain is given by K D = 1 log ρ : 2 ρ If, in particular, the domain D is the simply connected in C and the Gaussian curvature K D = 0 on D, then logρ = 0. Therefore ρ = je! j, where! is a holomorphic function on D. Thus the metric ρ is induced by the non-vanishing holomorphic function '(z) =e!(z) dened on the domain D. Since ρ 2 = '', a short computation yields 2ρρw = ' 0 ', and therefore 2(log ρ) w =(log') 0. It follows from (1.2) that f is '-harmonic, then (3.1) f zz + '0 2' f ff z f μz =0: Roughly speaking, '-harmonic maps arise if the curvature of the target is 0. Theorem 1.3 and (3.1) yield the following result. Theorem 3.1. Let f be a '-harmonic mapping of the unit disc U onto a C 2;ff Jordan domain D. If M = k(log ') 0 k1 < 1 and f is quasiconformal, then f has bounded partial derivatives. In particular, it is a Lipschitz mapping. Proof. The hypothesis M = k(log ') 0 k1 < 1, along with the equation (3.1) imply that the crucial hypothesis (1.5) of the main theorem is satised. Λ

9 INNER ESTIMATE AND Q.C. HARMONIC MAPS BETWEEN SMOOTH DOMAINS 125 Denition 3.2. A C 1 function χ satisfying the inequality j in a domain D is said to be approximately analytic in D with constant M. If a '-metric satises M 0 = k(log ') k1 < 1 on a domain D, then it is approximately analytic. This implies that j' 0 j»mj'j on D. Since j'j z»j' 0 j and 2ρ z ρ = j'j z, it follows that 2ρ z ρ = j'j z» j' 0 j» Mj'j = Mρ 2 on D. Thus the metric is approximately analytic in D with the constant M=2. The following theorem, concerning approximately analytic metrics, generalizes Theorem 3.1. Theorem 3.3. Let f be a C 2 ρ-harmonic mapping of the unit disc U onto the C 2;ff Jordan domain D. If the metric ρ is approximately analytic in D and f is quasiconformal, then f has bounded partial derivatives. In particular, it is a Lipschitz mapping. Theorem 3.3 follows directly from Theorem 1.3 (the main result), using the fact that the equation j = j@χj holds for all real functions χ. Denition 3.4. Let Ω be a Jordan domain and let z If r>0, then the set U z =Ω fw : jw zj <rg is called a neighborhood of z. Theorem 3.5 (Local version). Let f be a C 2 ρ-harmonic mapping of the unit disc U onto the C 2;ff Jordan domain D having a continuous extension f ~ to the boundary such that f ~ (@U) If f is quasiconformal in some neighborhood of a point z 0 2 T and the metric ρ is approximately analytic in some neighborhood of w 0 = f (z 0 ), then f has bounded partial derivatives and, in particular, is a Lipschitz mapping in a neighborhood of the point z 0. Proof. Let r > 0 such that f is q.c. in U 0 = D(z 0 ;r) U. Then fl 0 = f (T D(z 0 ;r)) is a C 2;ff Jordan arc containing w 0. Following the proof of Theorem 1.3, we obtain that the function f has bounded partial derivatives near the arc fl = f (T μ D(z 0 ;r=2)); and therefore, it must have bounded partial derivatives in some neighborhood of the point z 0. Λ The harmonic and q.c. mappings between Riemann surfaces The denition of a quasiconformal and harmonic mapping f : R! S between the Riemann surfaces R and S with the metrics % and ρ respectively is similar to the denition of those mappings between domains in the complex plane.

10 126 D. KALAJ AND M. MATELJEVIĆ If f is a harmonic mapping, then (3.2) Φdz 2 = ρ ff fz f μz dz 2 is a holomorphic quadratic differential on R. We call Φ the Hopf differential of f and write Φ=Hopf(f ). Lemma 3.6. Let (S 1 ;ρ 1 ) and (S 2 ;ρ 2 ) and (R; ρ) be three Riemann surfaces. Let g be an isometric transformation of the surface S 1 onto the surface S 2 : ρ 1 (!)jd!j 2 = ρ 2 (w)jdwj 2 ; w = g(!): Then f : R! S 1 is ρ 1 -harmonic if and only if g f f : R! S 2 is ρ 2 -harmonic. In particular, if g is an isometric self-mapping of S 1, then f is ρ 1 -harmonic if and only if g f f is ρ 1 -harmonic. Proof. If f is a harmonic map, then Φdz 2 = ρ f fpqdz 2 is a holomorphic quadratic differential in R, i.e., the mapping ρ ff pq is analytic if we consider it as a function of the parameter z = z( ), 2 R. Let! = f (z), F = g f f, P =(g f f )z and Q =(g f f ) μz. Then P = g 0 (!) p and Q = g 0 (!) q. Since ρ 1 (!) =ρ 2 (w)jg 0 (!)j 2 ; we obtain ρ 2 f FP μ Q = ρ 2 f g f f jg 0 (!)j 2 pq = ρ 1 f fpμq: Hence ' 1 =Hopf(g f f ) is a holomorphic differential, i.e., g f f is harmonic with respect to the metric ρ 2. Λ The remaining part of this section deals with the Riemann sphere. However, most of the arguments work for an arbitrary compact Riemann surface. We call the metric ρ dened on S 2 = C by ρjdzj 2 = 4jdzj2 (1 + jzj 2 ) 2 the spherical metric. The corresponding distance function is (3.3) ds(z; w) = p 2jz wj (1 + jzj 2 )(1 + jwj 2 ) ; ds(z;1) = p 2 (1 + jzj 2 ) The orientation preserving isometries of the Riemann sphere S 2 with respect to the spherical metric are given by Möbius transformations of the form (3.4) g(z) = az + b μa μ bz ; a; b 2 C, jaj2 + jbj 2 6=0: The Euler Lagrange equation for spherical harmonic mappings is (3.5) fzμz + 2 μ f 1+jfj 2 f z f μz =0: :

11 INNER ESTIMATE AND Q.C. HARMONIC MAPS BETWEEN SMOOTH DOMAINS 127 It is easy to verify that the spherical density is approximately analytic in C constant 1; more precisely, one has ρ = 2z 1+jzj 2 : ρ z with If f is a diffeomorphism of the Riemann sphere (or of a compact Riemann surface M) onto itself, then f is a quasi-isometry with respect to the corresponding metric and, consequently, is quasiconformal. It is natural to ask what can be said for harmonic q.c. diffeomorphisms dened in some sub-domain of the Riemann sphere. Using Theorem 3.3, Lemma 3.6 and the isometries dened by (3.4), we can prove the following Proposition 3.7. Let the domains Ω;D ρ C have C 1;ff and C 2;ff Jordan boundary on S 2 = C, respectively. Then any q.c. spherical harmonic diffeomorphism of Ω onto D is Lipschitz with respect to the spherical metric. 4 Representation of '-harmonic mappings If '(w 0 ) 6= 0, then there is a neighborhood V of w 0 and a branch of p ' in V R p R p such that f = '(z) dz is conformal on V. We refer to f = '(z) dz as the natural parameter on V dened by '. Theorem 4.1. If f is '-harmonic and ψ is conformal on the co-domain of f, then the mapping F = ψ f f satises the equation " # ψ (4.1) 00 (w) F zμz = ψ 0 (w) 2 (w) '0 F z F μz ; 2ψ 0 (w) '(w) where w = f (z). Proof. Since f is analytic, F z = ψ 0 (w) f z and F μz = ψ 0 (w) f μz. Hence F zμz = ψ 00 (w)f z f μz + ψ 0 (w)f zμz. On the other hand, f is '-harmonic, and therefore Now (4.1) follows easily. ' 0 f zz = 1 2 ' f f f z f μz : Observe that if ' =1, then the '-metric reduces to the Euclidean metric. So if f is a Euclidean harmonic mapping, then (4.2) F zμz = ψ00 ψ 0 2 F z F μz : Λ

12 128 D. KALAJ AND M. MATELJEVIĆ Corollary 4.2. Let ' be an analytic function such that there exists a branch of R p '(z) dz in some domain D. Iff :Ω! D is '-harmonic and Z p'(z) f = dz; then the mapping F = f f f is harmonic with respect to the Euclidean metric. Proof. It is easy to see that f 00 (w) (w) '0 f 0 (w) 2 2f 0 (w) =0: '(w) It follows from (4.1) that F zμz 0. Hence F is harmonic. Using (4.2) we obtain Corollary 4.3. Let h be a Euclidean harmonic mapping, and let ψ be conformal on the range of h; and let ' =((ψ 1 ) 0 ) 2. Then the mapping ^h = ψ f h is '-harmonic. Recall that if f is '-harmonic and f is the natural parameter dened by ', then the mapping F = f f f is Euclidean harmonic. Applying Theorem K (see the Introduction and also [12, Theorem 3.1]) to the C 1;ff domain D 0 = the f(d) and the Euclidean harmonic mapping F = f f f (note that f is not 1-1 in general), we can prove that Theorem 3.1 holds for more general domains. Theorem 4.4. Let f be a '-harmonic mapping of the C 1;ff domain Ω onto the C 1;ff Jordan domain D. IfM = k(log ') 0 k 1 < 1 and f is quasiconformal, then f has bounded partial derivatives. In particular, f is a Lipschitz mapping. R p Assume that '(z) 6= 0 and that the natural parameter f(z) = '(z) dz is well-dened on the domain D. Let f map D onto the convex domain D 0 = f(d). By the denition of the '-metric, we have Z p d(z; w) = j'( )jjd j: Since by the chain rule we obtain inf z;w2flρd p j'( )jjd j = jd(f( ))j; d(z; w) = fl Z inf jdοj; A;B2fl 0 ρd 0 fl 0 where A = f(z), B = f(w), ο = f( ), and D 0 = f(d). It follows that the segment [A; B] (which belongs to D 0,as D 0 is convex), is the curve that minimizes the previous functional. Hence d(z; w) =ja Bj = jf(z) f(w)j. We have proved the following Λ

13 INNER ESTIMATE AND Q.C. HARMONIC MAPS BETWEEN SMOOTH DOMAINS 129 Proposition 4.5. If D 0 = f(d) is convex, then f transforms the '-metric to the Euclidean metric; i.e., the distance function dened by the '-metric is given by the formula d(z; w) =jf(z) f(w)j : Example 4.6. Let ' 0 (w) =1=(w c 0 ) 2. Consider the harmonic maps between two domains Ω and D with respect to the metric density (4.3) ρ 0 (w) =j' 0 (w)j =1=jw c 0 j 2 ; w 2 D; on D, where c 0 62 D is a given point. If D 0 =log(d fc 0 g) is a convex domain, then the metric dened by (4.3) is d 0 (z; w) = log z c 0 w c 0 It is easy to verify that the conformal mappings (4.4) A(z) =c 0 + re iff(1 ") (z c 0 ) " ; r 2 R + ; " = ±1; : describe the orientation preserving isometries of the domain D ff = C nfc 0 + te iff ;t 2 R + g with respect to the metric d 0 given by (4.3). Let f be ' 0 -harmonic between simply connected domains Ω and D, where D ρ D ff for some ff. The natural parameter is f 0 (w) = ± log(w c 0 ). As an application of Corollary 4.2, we see that F (z) = log(f (z) c 0 ) is a harmonic function dened on Ω. Therefore, f (z) c 0 = e g0(z)+h0(z) = g 1 (z) h 1 (z) = which yields the representation (4.5) f (z) =g(z) +h(z) c 0 1 g(z) h(z); p c0 p 1 p g(z) c0 + 1 p h(z) ; c0 c0 where g and h are analytic mappings of Ω into C nfc 0 g. It is easy to see that the family of mappings dened by (4.5) is closed under transformations given by (4.4) (see Lemma 3.6). The above example provides the motivation for the following theorem. 1 Theorem 4.7. Let g and h be analytic functions and let f = g + h c 0 gh, c 0 6=0, be a diffeomorphism of the C 1; domain Ω onto the C 1;ff Jordan domain D such that c 0 2 C n D. Iff is q.c. mapping, then it has bounded partial derivatives, and the analytic functions g 0 and h 0 are bounded.

14 130 D. KALAJ AND M. MATELJEVIĆ Proof. The case c0 = 1 is proved by Theorem 4.4; therefore, we can assume that c0 6= 1. Put g1 = p c0 1 p c 0 g and h 1 = p c0 + 1 p c 0 h: Then f c0 = g1 h1. Since f (z) 6= c0 it follows that h1(z) 6= 0and g1(z) 6= 0. The mapping F = log(f c0), which can be written as F = log g1 + log h1 on Ω, isa harmonic mapping of Ω onto C 1;ff domain D 0 =log(d c0). Then Theorem 4.4 implies that there exists a constant M such that (4.6) jh 0 1=h1j 2 + jg 0 1=g1j 2 <M: Thus (log h1) 0 is bounded on Ω, and consequently log h1 has a continuous extension to Ω. Therefore, h1 has a continuous and non-vanishing extension to Ω. The same holds for g1. The inequality (4.6) implies that h 0 1 and g1 0 are bounded mappings. Thus h 0 and g 0 are bounded. Λ Example 4.8. A harmonic mapping u with respect to the hyperbolic metric on the unit disk satises the equation u zμz + 2μu 1 juj 2 u z uμz =0: As far as we know, this equation cannot be solved using known methods of PDE. However, we can produce some examples. More precisely, we characterize real hyperbolic harmonic mappings. Let '1(w) =4=(1 w 2 ) 2 : Using the natural parameter, i.e., a branch of f1(w) =log 1+w 1 w = 2 tanh 1 (w), one can verify that f is '1-harmonic if and only if f =tanhg, where g is Euclidean harmonic. Since the metric ρ = j'(w)j coincides with the Poincaré metric =4=(1 jwj 2 ) 2 ; for real w we obtain that f is real -harmonic (hyperbolic harmonic) if and only if f =tanhg, where g is real Euclidean harmonic. Since the mappings w = e z a i' ; (jaj < 1); 1 μaz are the isometries of the Poincaré disc, it follows from Lemma 3.6 that if h is real harmonic dened on some domain Ω, then the function i' tanh(h(z)) a (4.7) w = e (jaj < 1) 1 μa tanh(h(z))

15 INNER ESTIMATE AND Q.C. HARMONIC MAPS BETWEEN SMOOTH DOMAINS 131 is harmonic with respect to the hyperbolic metric. Note that the mappings given by (4.7) map Ω into circular arcs orthogonal to the unit circle T. Moreover, if a circle S orthogonal to the unit circle T is given, we can use (4.7) to describe all -harmonic mappings between Ω and S U. Acknowledgment. We thank the referee for useful comments and suggestions related to this paper. A part of this manuscript was communicated by the second author at the X-th Romanian Finnish Seminar, August 14 19, 2005, Cluj-Napoca, Romania and at the Helsinki Turku Seminar (during the visit of the second author to the Universities of Helsinki and Turku in October 2005). Participants of those seminars gave useful comments; in particular, it was suggested by Olli Martio that Theorem MK1-2 may have a several dimension analogue (details about this will appear in [21]). Finally, we thank Edgar Reich, Matti Vuorinen, Aleksander Poleksic and Stephen Taylor for their interest in this paper and for their helpful advice concerning the language. REFERENCES [1] L. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, Princeton, N.J., [2] I. Anić, V. Marković and M. Mateljević, Uniformly bounded maximal '-discs and Bers space and harmonic maps, Proc. Amer. Math. Soc. 128 (2000), [3] A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), [4] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. A I Math. 9 (1984), [5] P. Duren, Harmonic Mappings in the Plane, Cambridge Univ. Press, [6] Z-C. Han, Remarks on the geometric behavior of harmonic maps between surfaces, Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), A. K. Peters, Wellesley, MA, 1996, pp [7] G. M. Goluzin, Geometric Function Theory, Nauka, Moscow, 1966 (Russian). [8] E. Heinz, On certain nonlinear elliptic differential equations and univalent mappings, J. Analyse Math. 5 (1956/57), [9] D. Kalaj, Harmonic Functions and Harmonic Quasiconformal Mappings Between Convex Domains, Thesis, Beograd, [10] D. Kalaj, On harmonic diffeomorphisms of the unit disc onto a convex domain, Complex Var. Theory Appl. 48 (2003), [11] D. Kalaj, Quasiconformal harmonic functions between convex domains, Publ. Inst. Math. (Beograd) (N. S.), 76(90) (2004), [12] D. Kalaj, Quasiconformal harmonic functions between Jordan domains, preprint. [13] D. Kalaj and M. Pavlović, Boundary correspondence under harmonic quasiconformal homeomorphisms of a half-plane, Ann. Acad. Sci. Fenn. Math. 30 (2005),

16 132 D. KALAJ AND M. MATELJEVIĆ [14] O. Kellogg, On the derivatives of harmonic functions on the boundary, Trans. Amer. Math. Soc. 33 (1931), [15] M. Knežević and M. Mateljević, On q. c. and harmonic mapping and the quasi-isometry, preprint. [16] P. Li, L. Tam and J. Wang, Harmonic diffeomorphisms between hyperbolic Hadamard manifolds, to appear in J. Geom. Anal. [17] O. Martio, On Harmonic Quasiconformal Mappings, Ann. Acad. Sci. Fenn. Ser. A I 425 (1968). [18] M. Mateljević, Note on Schwarz lemma, curvature and distance, Zb. Rad. (Kragujevac) 16 (1994), [19] M. Mateljević, Ahlfors-Schwarz lemma and curvature, Kragujevac J. Math. 25 (2003), [20] M. Mateljević, Dirichlet s principle, distortion and related problems for harmonic mappings, Publ. Inst. Math. (Belgrad) (N. S.) 75(89), (2004), [21] M. Mateljević, Distortion of harmonic functions and harmonic quasiconformal quasi-isometry, submitted to Proceedings of the X-th Romanian Finnish Seminar, August 14-19, 2005, Cluj- Napoca. [22] M. Pavlović, Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disc, Ann. Acad. Sci. Fenn. 27 (2002), [23] C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, [24] Y. Shi, L. Tam and T. Wan, Harmonic maps on hyperbolic spaces with singular boundary values, J. Diff. Geom. 51 (1999), [25] L. Tam and T. Wan, Harmonic diffeomorphisms into Cartan Hadamard surfaces with prescribed Hopf differentials, Comm. Anal. Geom. 4 (1994), [26] L. Tam and T. Wan, Quasiconformal harmonic diffeomorphism and universal Teichmüler space, J. Diff. Geom. 42 (1995), [27] L. Tam and T. Wan, On quasiconformal harmonic maps, Pacic J. Math. 182 (1998), [28] T. Wan, Constant mean curvature surface, harmonic maps, and universal Teichmüller space, J. Diff. Geom. 35 (1992), [29] A. Zygmund, Trigonometric Series I, Cambrige University Press, David Kalaj FACULTY OF NATURAL SCIENCES AND MATHEMATICS UNIVERSITY OF MONTENEGRO CETINJSKI PUT B.B PODGORICA, MONTENEGRO davidk@cg.yu Miodrag Mateljević FACULTY OF MATHEMATICS, STUDENTSKI TRG 16 UNIVERSITY OF BELGRADE BELGRADE, SERBIA miodrag@matf.bg.ac.yu (Received October 9, 2005 and in revised form January 18, 2006)