Theorem 1. Suppose that is a Fuchsian group of the second kind. Then S( ) n J 6= and J( ) n T 6= : Corollary. When is of the second kind, the Bers con
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1 ON THE BERS CONJECTURE FOR FUCHSIAN GROUPS OF THE SECOND KIND DEDICATED TO PROFESSOR TATSUO FUJI'I'E ON HIS SIXTIETH BIRTHDAY Toshiyuki Sugawa x1. Introduction. Supposebthat D is a simply connected domain of hyperbolic type in the Riemann sphere C = C [ f1g: Then the Poincare metric D in D is dened by D (z) = g (z) 1 g(z) ; z 2 D; 2 where g is any conformal mapping of D onto the unit disk 1 = fz : z < 1g. B 2 (D) will denote the Banach space consisting of all holomorphic functions ' in D such that the norm k'k D = sup '(z) D (z) 2 z2d is nite. If f is a locally univalent meromorphic function, the Schwarzian derivative off is given by We set after Flinn [6] S f = f f 1 2 f f 2 : S = fs f : f is conformal in 1g; J = fs f 2 S : f(1) is a Jordan domain g; T = fs f 2 S : f(1) is a quasidisk g: T is called the universal Teichmuller space. It is known that T J S B 2 (1);T is open, S is closed and T =IntS (see [7], [9]). Let be a Fuchsian group acting on 1 and B 2 (1;) denote the closed subspace of B 2 (1) : f' 2 B 2 (1):(' ) 1 ( ) 2 = ' for all 2 g: Further we set S( )=S \ B 2 (1;);J( )=J \ B 2 (1;);T( )=T \ B 2 (1;): Then T ( ) coincides with the Bers embedding of the Teichmuller space of (see [3]). Bers conectured that S = T i.e., S(1) = T (1) in [2]. Generalizing this conecture, by the Bers conecture fo we shall mean that S( )=T ( ): In [8], Gehring showed that the Bers conecture is false i.e., S n T 6= ; in fact essentially he showed that S n J 6= : Moreover Flinn proved that J n T 6= in [6] (see the next section for the details). Our main purpose in this paper is to show the following Typeset by AMS-TEX 1
2 Theorem 1. Suppose that is a Fuchsian group of the second kind. Then S( ) n J 6= and J( ) n T 6= : Corollary. When is of the second kind, the Bers conecture fo is false. In other words, S( ) % T ( ): On the other hand, the author does not know whether the Bers conecture for Fuchsian groups of the rst kind is true or not. This paper is organized as follows. In x2 weintroduce simply connected domains which are constructed by Gehring and Flinn and quote two results from Flinn [6] for later use. Gehring's (or Flinn's) domain is essentially obtained by removing a spiral (respectively, countable sequence of closed Jordan regions approximating a spiral) from a disk so that the boundary of the domain is adequately irregular. For a given Fuchsian group of the second kind, in x3 we construct a -invariant simply connected domain 1 in 1 which contains the Gehring's or Flinn's domain adacent to a free side of the Dirichlet fundamental region of : Let F :1! 1 be a Riemann mapping function of 1 ; then S F 2 S(G) where G = F 1 F is a Fuchsian group. While it turns out later that S F =2 J (respectively, S F =2 T ) and that G is qc(=quasiconformally) equivalent to (Lemma 4, Corollary), these facts need not imply that S( ) n J 6= (respectively, J( ) n T 6= ): Now we consider to deform 1 by an appropriate qc mapping so that the Schwarzian derivative as above belongs to S( ): In x4, such a deformation is presented and we state a slightly general result (Theorem 2) which contains Theorem 1. x5 isdevoted to the proof of Theorem 2. x2. Gehring's and Flinn's construction. Fix a 2 (; 1 ) and set a closed 8 Jordan arc a = f6ie (ai)t : t 2 [; 1)g[fg: Theorem A (Gehring [8]). Let F :1! b C n a beariemann mapping function of b C n a ; then S F 2 S n J: We set A = fx iy : y>1g[fx iy : x>4;y >1g and D 1 = A n a : As we shall nd later, the following theorem also holds. Theorem A'. Let F : 1! D 1 beariemann mapping function of D 1 ; then S F 2 S n J: For each positive integer m2 N,we set m =( 8 )m ; m = e 2am ;E m = R m [P m where P m = fe i z : z 2 a ; m m g[fz : z m g; R m = fx iy : x sin m ; 1 y cos m gn1: T 1 Then each E m is a closed Jordan region, E 1 E and m=1 E m = a : Let V denote the translation V (z) =z 8 and b S 1 set D 2 = A n m=1 V m (E m ): One can easily see that D 2 is a Jordan domain in C: The next theorem is essentially due to Flinn. 2
3 Theorem B (Flinn [6, Theorem 2]). Let F :1! D 2 beariemann mapping function of D 2 ; then S F 2 J n T: Theorems A' and B are direct conclusions of the following Lemmas 1 and 2, respectively. Let 1 = fz = e (ai)t : t 2 (; 1)g; 2 = fz : z 2 1 g: Then 1 and 2 are logarithmic spirals in D 2 which converge onto the point from opposite sides of a : Lemma 1 (cf. Flinn [6, Lemma 2]). There exists a constant 1 > with the following property. If f is conformal in D 1 with ks f k D1 1 ; then lim f(z) = lim f(z): 1 3z! 2 3z! In particular, f(d 1 ) is not a Jordan domain. Let be the subarc fx iy 1 : 4 <x<1g 1 : We note that if f : R 1! R 2 is a conformal mapping of a Jordan domain R 1 onto another Jordan domain R 2 ; then f is uniquely extended to a homeomorphism ~ f : R 1! R 2 : Lemma 2 (cf. Flinn [6, Proof of Theorem 2]). There exists a constant 2 > with the following property. If f is conformal in D 2 with ks f k D2 2 and if f(d 2 ) is a Jordan domain, then ~ f() is not a quasiarc. Remark. In Thorems A' and B, we can replace the domain A by a half plane fx iy : y>1g: x3. Construction of group invariant domains. Let be an arbitrary Fuchsian group of the second kind acting on the unit disk 1: In this section we construct - invariant simply connected domains which have the same property as the Gehring's or Flinn's domain. Since is of theb second kind, ( ) 6= where ( )is the region of discontinuity of in C: Now we pick a suciently small disk Y in ( ) whose boundary is orthogonal so that no two distinct points of Y are -equivalent. Let beamobius transformation such that (Y ) = 1 and that (Y )=1 where Y = Y \ 1 and 1 = fz 2 1:Imz>g: Fix ;r 1 2 (; 1) such that r 1 < : Let 1 r = fz 2 1:z <rg; 1 =1 r r \ 1 and Y r = 1 (1 ) r for r 2 (; 1): Let M(1) be the space of Beltrami coecients supported in 1 : f 2 L 1 (1) : kk 1 < 1g: For 2 M(1);w will denote the normalized -conformal self-mapping of 1 which xes three points 1;i;1 We set M Y (1) k = f 2 L 1 (1) : =ony r r and kk 1 <kg; for k 2 (; 1]: Notice that w is conformal in Y for 2 M Y (1) k : Lemma 3. Let = minf 1 ; 2 g where 1 and 2 are asinlemmas 1 and 2, respectively, then there exists a constant k 2 (; 1] such that for any 2 M Y (1) k the following is valid: ks w k Y r :
4 Proof. We can extend w to a qc homeomorphism of b C by the rule w (z) =1=w (1=z): It is well known that w (z) converges to z uniformly on each compact set in C as kk 1! (see [1], for example). Since w is conformal in Y r = 1 (1 r ) for 2 M Y (1) 1 and Y r r 1 is relatively compact in Y r ;S w converges to uniformly on Y r 1 as kk 1! ; in particular ks w k Y converges to as kk 1! : 3 r 1 For each " 2 (;r 1 )wechoose a = " 2Mob such that (b R)=fx i : x 2 Rg [ f1g and (1 " ) A: [ Construction 1 (Gehring type). We set 1 1 =1 1 (") =1n (( ) 1 ( a )): 2 Construction 2 (Flinn type). [ We set 1 2 =1 2(") =1n 2 (( ) 1 ( S 1 m=1 V m (E m ))): S We note that ( ) 1 ( a ) Y ; that ( ) 1 1 ( m=1 V m (E m )) Y and that ((Y )) 2 is a disoint family. Therefore 1 is a -invariant simply connected domain contained in 1 for =1; 2; furthermore 1 2 is a Jordan domain. If we let F :1! 1 be a Riemann mapping function and set G = F 1 F which isa subgroup of Mob acting discontinuously on 1 i.e., a Fuchsian group acting on 1; then S F1 2 S(G 1 ) and S F2 2 J(G 2 ): Now we state a lemma which guarantees that G is qc equivalent to: Lemma 4. Let k 2 (; 1] be asinlemma 3. For suciently small " 2 (;r 1 ); there exists a qc mapping h of 1 =1 (") onto 1 with the following properties for =1; 2. (1) k(h )k 1 <kwhere (h ) is the Beltrami coecient of h ; (2) h is conformal in 1 \ Y ; (3) h = h for all 2 : Because the qc mapping f = h F :1! 1 deforms G into ; we have the following Corollary. G is qc equivalent to : Lemma 4 is obtained in an obvious way by the following Lemma 5. For suciently small " 2 (;r 1 ); there exists a qc mapping H : (Y )! 1 with the following properties for =1; 2. (1) k(h )k 1 <kwhere (H ) is the Beltrami coecient of H ; (2) H is conformal in (Y ) \ 1 ; (3) H = identity n [1; 1]; where Y = Y (") =Y \ 1 ("): Proof of Lemma 5. Let H (") : (Y (")) \ 1! 1 be the conformal mapping which xes three points ; i; : Noting that 1 n 1 " (Y (")); we nd that 4
5 g H (") 1 can be extended to a conformal mapping H(") in fz : "<z <r 2 ="g n1 " by the reection principle. g Let 2 " :(;)! (;) be the mapping dened by the rule e i2 "() = H(") ( e i ) for all 2 (;); then 2 " is a smooth mapping such that 2"() =H (") (e i ): Now we set H (") (re i )=re i(t(1t)2 "()) for r 2 [ ; 1]; 2 (;); where r = t (1t) ; then g this extended H (") has the properties (2) and (3). On the other hand, clearly H(") (z) converges to z uniformly on each compact set in C 3 = C nfg as "! ; hence 2 " ();2 "() uniformly converges to ; 1as"! ; respectively. Therefore the ( explicit expression (rr )(2 " ()1)(2 "())ri (H (") )(re i 2(1r ) = ; r 2 ( )(r )(2 " ()1)(2 "())ri ; 1]; ; r 2 (; ) shows that k(h (") )k 1! as"! ; so for suciently small " 2 (;r 1 ); H = H (") : (Y )! 1 has the properties (1), (2) and (3). 3 Henceforth we xan" 2 (;r 1 ) for which Lemma 4 holds. x4. Deformations by partially conformal qc mappings. In this section we present a method to construct the family of group-invariant domains which includes desired one. Let M(1;) be the space of Beltrami coecients fo with support in 1 i.e., the subset of L 1 (1) consisting of all 2 L 1 (1) with kk 1 < 1 and ( ) 1 = = for all 2 : Set D = w (1 ) for 2 M(1;): If denotes the Fuchsian group w (w ) 1 acting on 1; then D is a -invariant simply connected domain whose boundary is homeomophic to the 1 's. We take a Riemann mapping function F :1!D and set G = (F )1 F and ' = S F : Since acts discontinuously on D ;G acts also discontinuously on 1; hence G is a Fuchsian group. And clearly ' 1 2 S(G 1 ) and ' 2 2 J(G 1 ): Because the logarithmic spiral a is a quasiarc, the general qc mapping w may unfasten the spirals removed, thus we must restrict Beltrami coecients to be considered on a certain class of M(1;): In this article we only consider M Y r (1;) k = M Y r (1) k \ M(1;) = f 2 M(1;): =ony and kk 1 <kg: Since w is conformal in Y for 2 M Y (1;) k ; it is expected that the spirals are but slightly deformed by w : In fact, we have the following result for this class which is proved in the rest of this paper. 5
6 Theorem 2. Let k be asinlemma 3. For 2 M Y (1;) k ; we have ' 1 2 S(G 1 ) n J; ' 2 2 J(G 2 ) n T: Theorem 2 and Lemma 4 prove Theorem 1. Proof of Theorem 1. Let h be as in Lemma 4. We set = (h ); on 1 ; ; on 1 n 1 ; where (h ) is the Beltrami coecient ofh and set D = w (1 ). Notice that 2 M Y (1;) k : Since h and w 1 have the same Beltrami coecient, w h 1 :1!D is a Riemann mapping function of D : Therefore we can take w h1 as F : By denition, G = h (w )1 w h1 = h h 1 = : By virture of Theorem 2, ' S( ) n J;' J( ) n T; hence Theorem 1 is proved. 3 Remark. The family (' ) 2M Y (1; ) k is ample in a certain sense. We now explain this in the following. We recall that F :1! 1 is a Riemann mapping function of 1 and G = F 1 F isafuchsian group acting on 1: In this paragraph we assume that is non-elementary and choose F so that 1;i;1 2 3(G )= bcn (G ): Let F 3 : M(1;)! M(1;G ) be the pullback of Beltrami coecients by F ; namely F 3 () is the Beltrami coecient of the qc mapping w F for 2 M(1;): Since w F 3 () and w F have the same Beltrami coecient, we can choose w F (w F 3 () ) 1 :1!D as the Riemann mapping function F ; then we have G = 3 wf () G (w F 3 () ) 1 : Generally, for 2 M(1;G ) the group isomorphism g 7! w g(w ) 1 (g 2 G ) determines an element of the reduced Teichmuller space T # (G )ofg (see, for example, Earle [4], [5], Nag [1]). Let this point int # (G ) be denoted by 8 # (): It turns out that 8 # (M K (1;G ) k ) is a neighborhood of 8 # () in T # (G ) for any k 2 (; 1] and any measurable set K 1 such that p(k) is relatively compact in the double (G )=G where p : (G )! (G )=G is the canonical proection. (For example, combine [11, Corollay 2] with [4]. This fact was pointed out to the author by H.Ohtake.) On the other hand F 3 (M Y relatively compact in (G )=G ; hence 8 # (F 3 (M Y (1;) k )=M 1 F (Y r )(1;G ) k and p(f 1 (Y r )) is (1;) k )) is a neighborhood of 8 # (): In other words, qc deformations G! G (g 7! wf 3 () g(w F 3 () ) 1 ) for 2 M Y (1;) k cover a neighborhood of the identity mapping G! G in T # (G ): The above proof of Theorem 1 shows virtually that there exists an isomorphism G! which belongs to 8 # (F 3 (M Y r (1;) k )): 6
7 x5. Proof of Theorem 2. Proof of the rst part. ' 1 =2 J: By the same argument in [6] or [8], it is sucient toprove the following Claim 1. There exists a constant > with the following property. If f is conformal in D 1 with ks f k D ; then f(d 1 1 ) is not a Jordan domain. Proof of Claim 1. We set = 1 =2 where 1 is as in Lemma 1. Suppose that f is conformal in D 1 with ks f k D : Set g = f w ( ) 1 D1 ; 1 = w ( ) 1 ( ) for =1; 2 and w = w ( ) 1 (): Since Lemma 1 implies that ks g k D1 = ks fw k () 1 (D 1 ) ks f k D 1 ks w k Y r 1 Thus f(d 1 ) is not a Jordan domain. 3 lim f(w) = lim f(w): 1 3w!w 2 3w!w Proof of the second part. ' 2 =2 T: Similary it is sucient to prove the following 1 ; Claim 2. There exists a constant > with the following property. If f is conformal in D 2 with ks f k D ; then f(d 2 2 ) is not a quasidisk. Proof of Claim 2. We set = 2 =2 where 2 is as in Lemma 2. Suppose that f is conformal in D 2 with ks fk D : Further suppose that f(d 2 2 ) is a Jordan domain. We shall show 2 )isnot a quasicircle. Set g = f w ( ) 1 D2 : Since ks g k D2 ks f k D ks w k 2 Y 2 and g(d 2 )isajordan domain, Lemma 2 r 1 produces that ~g() is not a quasiarc. 2 ) is not a quasicircle because 2 ): 3 Acknowledgement. The author would like to thank Prof. Masahiko Taniguchi for giving helpful suggestions, which greatly simplied his previous argument. He also thanks Prof. Hiromi Ohtake for valuable informations. Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 66, Japan References 1. L.V.Ahlfors and L.Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. 72 (196), 385{ L.Bers, On boundaries of Teichmuller spaces and on kleinian groups I, Ann. of Math. 91 (197), 57{6. 3. A.Douady and C.J.Earle, Conformally natural extension of homeomorphism of the circle, Acta Math. 157 (1986), 23{ C.J.Earle, Teichmuller spaces of groups of the second kind, Acta Math. 112 (1964), 91{97. 5., Reduced Teichmuller spaces, Trans. Amer. Math. Soc. 126 (1967), 54{63. 7
8 6. B.B.Flinn, Jordan domains and the universal Teichmuller space, Trans. Amer. Math. Soc. 282 (1984), 63{ F.W.Gehring, Univalent functions and the universal Teichmuller space, Comment. Math. Helv. 52 (1977), 561{ , Spirals and the universal Teichmuller space, Acta Math. 141 (1978), 99{ O.Lehto, Univalent Functions and Teichmuller Spaces, Springer-Verlag, S.Nag, The Complex Analytic Theory of Teichmuller Spaces, Wiley, H.Ohtake, On the deformation of Fuchsian groups by quasiconformal mappings with partially vanishing Beltrami coecients, J. Math. Kyoto Univ. 29 (1989), 69{9. 8
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