Research Article On Geodesic Segments in the Infinitesimal Asymptotic Teichmüller Spaces

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1 Joural of Fuctio Spaces Volume 2015, Article ID , 7 pages esearch Article O Geodesic Segmets i the Ifiitesimal Asymptotic Teichmüller Spaces Ya Wu, 1,2 Yi Qi, 1 ad Zuwei Fu 2 1 LMIB ad School of Mathematics ad Systems Sciece, Beihag Uiversity, Beijig , Chia 2 Departmet of Mathematics, Liyi Uiversity, Liyi , Chia Correspodece should be addressed to Zuwei Fu; wfu@mail.bu.edu.c eceived 27 October 2015; Accepted 3 December 2015 Academic Editor: Staislav Hecl Copyright 2015 Ya Wu et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Let AZ() be the ifiitesimal asymptotic Teichmüller space of a iema surface of ifiite type. It is kow that AZ() is the quotiet Baach space of the ifiitesimal Teichmüller space Z(),where Z() is the dual space of itegrable quadratic differetials. The purpose of this paper is to study the ouiqueess of geodesic segmet joiig two poits i AZ(). We prove that there exist ifiitely may geodesic segmets betwee the basepoit ad every osubstatial poit i the uiversal ifiitesimal asymptotic Teichmüller space AZ(D) bycostructiga special degeeratig sequece. 1. Itroductio Let be a hyperbolic iema surface, that is, a iema surface with uiversal coverig surface which is coformally equivalet to the ope uit disk D. DeotebyL () the Baach space of Beltrami differetials μ = μ()d/d o X with fiite L -orms. Let Q() be the space of itegrable holomorphic quadratic differetials =()d 2 o with L 1 -orms: = () dx dy < +. (1) Deote by Q 1 () the uit sphere of Q() ad by Q d () the set of all degeeratig sequeces of Q(). Asequeceof quadratic differetials { } i Q() is said to be a degeeratig sequece if =1ad 0locally uiformly o as. Two elemets μ, ] L () arecalledifiitesimally Teichmüller equivalet, if μ = ] Q(). (2) We deote by [μ] the ifiitesimal Teichmüller equivalece class of μ. TheifiitesimalTeichmüller space Z () fl {[μ] :μ L ()} (3) is the set of all ifiitesimal Teichmüller equivalece classes of μ s i L ().Thepoit[0] is called the basepoit of Z(). A Beltrami differetial μ L () is said to be vaishig at ifiity, if, for each ε>0,thereisacompactsete i such that μ E <ε.deotebyl 0 () the set of all such vaishig μ s ad by Z 0 () fl {[μ] Z() :μ L 0 } (4) the set of all ifiitesimal Teichmüller equivalece classes of μ s i L 0 (). Two elemets μ, ] L () are called ifiitesimally asymptotically equivalet, if there exists σ L 0 () such that μ = ]+ σ Q (). (5) We deote by [[μ]] the ifiitesimal asymptotic equivalece class of μ. TheifiitesimalasymptoticTeichmüller space AZ () fl {[[μ]] :μ L ()} (6) is the set of all ifiitesimal asymptotic equivalece classes of μ s i L (). Thepoit[[0]] is called the basepoit of AZ(). It is kow that Z() ad AZ() are the taget spaces to the classical Teichmüller space ad asymptotic Teichmüller

2 2 Joural of Fuctio Spaces space at their basepoit, respectively (see [1]). For further results ad properties about Teichmüller theory,we refer to the papers [1 5] ad the books [6 8]. The otio of geodesics plays a importat role i the study of the geometry of Teichmüller theory. A geodesic i a metric space is a cotiuous curve such that for ay subarc its legth is equal to the distace betwee its two edpoits. The existece ad uiqueess of geodesic betwee two poits i various spaces have bee discussed for a log time (see [6, 9 15]). Give two poits i ifiitesimal Teichmüller space Z(), it is show i [6] that there exists at least oe geodesic joiig them. Furthermore, it is proved i [14] that there exists precisely oe geodesic segmet joiig the basepoit [0] ad [μ] i Z() if ad oly if [μ] cotais a uiquely extremal Beltrami coefficiet with costat modulus. The existece of geodesic joiig two give poits i AZ() hasbeeprovedi[1,6];however,theuiqueess of geodesic is ukow. I this paper, we will prove the ouiqueess of geodesics i the uiversal ifiitesimal asymptotic Teichmüller space AZ(D) by costructig a special degeeratig sequece { } i Q(D). The structure of this paper is as follows. Sectio 2 is devoted to settig up the otatios ad some results we eed. I Sectio 3, a special sequece { } degeeratig towards a boudary poit of the uit circle is costructed. I Sectio 4, it is proved that there are ifiitely may geodesics joiig [[μ]] with the basepoit i AZ(D) whe [[μ]] is of a osubstatial poit usig the importat lemma (Lemma 2) adthecostructedsequeceitheprevioussectio. 2. Preiaries I this sectio, we recall some otios ad basic results from the Teichmüller theory. For more details, we refer to the book [6] ad the paper [1]. By the Hah-Baach ad ies represetatio theorem, every elemet V i the dual space Q () of the Baach space Q() of all itegrable holomorphic quadratic differetials ca be represeted as V()= μ, Q (), (7) where μ is a Beltrami differetial i L (). Sothereis atural oe-to-oe correspodece betwee the ifiitesimal Teichmüller space Z() ad the dual space Q (). Thus, i what follows, the ifiitesimal Teichmüller equivalece classesofbeltramidifferetialso ad complex liear fuctioals o Q() areusedforpoitsofz() alterately. For every V Z(), the ifiitesimal extremal dilatatio ad the ifiitesimal boudary dilatatio of V are defied as V = [μ] = if { μ :μrepresets V}, (8) b (V) =b([μ] ) = if {b (μ) :μrepresets V}, (9) respectively, where b (μ) = if { μ E :Eis a compact subset of }. (10) μ is called ifiitesimal extremal if μ = V.Itisshow i [6] that Z() is a Baach space with the ifiitesimal Teichmüller orm V = sup Q 1 () μ = if {b (μ): μ represets V}. (11) The ifiitesimal Teichmüller distace betwee two poits V 1 ad V 2 i Z() is defied i [14] d(v 1,V 2 )= V 1 V 2 = sup Q 1 () (μ 1 μ 2 ), (12) where μ 1 ad μ 2 represet V 1 ad V 2,respectively. Let P : Z() AZ(); V V be the quotiet mappig from the taget space Z() to the taget space AZ(). V 1 ad V 2 i Z() represet the same poit V i AZ() if V 1 V 2 Z 0 (). Thus, the ifiitesimally asymptotic equivalece classes [[μ]] of Beltrami differetials are i oeto-oe correspodece with the elemets V of AZ() ad μ represets V. For ay V AZ(), we defie the quotiet orm o the quotiet space AZ() as b( V) = b ([[μ]] )=if {b (μ): μ represets V}. (13) Itiskowi[6]thatb( V) = b(v). μ is called ifiitesimally asymptotically extremal if b (μ) = b( V). Furthermore, AZ() is a Baach space with the stadard semiorm (see [6]) b( V) = sup { } Q d () = sup { } Q d () sup V( ) sup (14) μ, where μ represets V.Iparticular,if V Z 0 (),itholds b ( V) = sup { } Q d () sup μ =0. (15) The ifiitesimal asymptotic Teichmüller distace betwee two poits V 1 ad V 2 i AZ() is defied as d( V 1, V 2 )=b( V 1 V 2 ) = sup { } Q d () sup (μ ]), (16) where μ ad ] represet V 1 ad V 2,respectively. For ay V Z(), μ L () is a ifiitesimal extremal represetative of V if ad oly if it has a so-called Hamilto sequece, amely, a sequece { } Q 1 (),suchthat μ = μ. (17) Similarly, for ay V AZ(), μ L () is a ifiitesimal asymptotically extremal represetative of V [1, 6] if ad oly

3 Joural of Fuctio Spaces 3 if there exists a asymptotic Hamilto sequece of μ,amely, a degeeratig sequece { } Q d (),suchthat μ =b (μ). (18) Let D ={: <1}betheuit disk i the exteded complex plae Ĉ ad let D be the uit circle. I the followig part,wecosidersomeresultsabouttheifiitesimallocal boudary dilatatio of V i the taget space Z(D) to the uiversal Teichmüller space. Set p Dad U r (p) = D { : p < r}. Foray V Z(D), the ifiitesimal local boudary dilatatio of V at p is defied as b p (V) =b p ([μ] )=if {b p (μ): μ represets V}, (19) where b p (μ) = if r For ay V AZ(D), we defie { μ () U r (p) : U r (p)}. (20) b p ( V) = b p ([[μ]] )=if {b p (μ): μ represets V}, (21) ad the b p (V) = b p ( V).ItisprovedbyLakic[16]that b( V) = max p D b p ( V). (22) Apoitp D with b p ( V) = b( V) is said to be a ifiitesimal substatial boudary poit for V. V (or [[μ]] ) is called a ifiitesimal substatial poit i AZ(D),ifevery p D is a ifiitesimal substatial boudary poit for V (or [[μ]] ); otherwise, V (or [[μ]] )iscalledaifiitesimal osubstatial poit. The followig lemma ca be obtaied i [17] by Fehlma ad Saka. Lemma 1. For ay V Z(D), letμ L (D) be a ifiitesimal extremal represetative of V. Suppose there is a poit p D which is ot a substatial boudary poit of V. The, there is a ope iterval I D, p I,adadomai S I ={ D =0, I}, (23) such that, for ay degeeratig Hamilto sequece {ψ } of μ, oe has ψ dx dy = 0. (24) S I 3. Costructig a Special Sequece Degeeratig towards a Boudary Poit I this sectio, we will costruct a special sequece degeeratig towards a boudary poit p D.Themethodused here is similar to that i [18] while the sequece degeerates towardsthewholeboudaryithispaper. Let p D ad U r ={ D : p <r}. A degeeratig sequece { } Q d (D) is said to degeerate towards p if, for ay eighbourhood U r of p with r>0, () U r (p) dx dy = 1. (25) The, for ay ε>0ad r>0, there exists a positive iteger N,suchthat 1 ε<u r () dx dy < 1 (26) holds for every >N.Sice = 1, N, there exists a positive umber r 1 <1satisfyig D\D r1 1 > (27) By the defiitio of degeeratig sequece ad (26), there exists k2 such that k 2 < 1 2 2, for 1 r 1, (28) D\U r1 k 2 < (29) Choose a positive umber r 2 <r 1 such that D\U r2 k 2 > , (30) U r2 k 1 < 1 2 2, (31) where k 1 =1. It follows from (29) ad (30) that U r1 \U r2 k 2 > (32) By iductio, we obtai a subsequece { k } of { k } ad a positive umber sequece {r } with r < r 1 ad r =0such that k < 1 2, i ( 1 r ), (33) k U r 1 \U >1 1 r 2, (34) U r k j < 1 2, j=1,2,..., 1, (35) where = 2, 3,.... Without loss of geerality, from ow o, we write istead of k for simplicity. Let = =1. (36)

4 4 Joural of Fuctio Spaces From (33), it is easy to see that this series is uiformly coverget i every compact subset of D.So is a holomorphic quadratic differetial o D. Notig that =1ad by (34), we get that is, 1 U r 2, D\U r ; (37) =O(2 ),, U r D\U r 1 =O(2 ),. Moreover, by the secod formula of (37), we have (38) j 1 D\U r 2 j, j +1. (39) By simple calculatio, it follows from (35), (37), ad (39) that The, U r 1 \U r 1 < j=1 + 1 j=1 j U r 1 \U r j=+1 j U r 1 \U r U j + j r 1 j=+1 D\U r 1 1 < j=1 j=+1 1 < 2j 2 1. (40) =O(2 ( 1) ),. (41) U r 1 \U r Furthermore, from (34) ad (41), we have =1+O(2 ( 1) ),. (42) U r 1 \U r 4. Nouiqueess of Geodesics Joiig Every Ifiitesimal Nosubstatial Poit with the Basepoit i AZ(D) For every V AZ(), itiskowi[1]thatthereexistsa represetative μ such that μ =b( V).The μ =b( V) V μ (43) meas μ is a ifiitesimal extremal represetative of V,ad μ =b( V) b (μ) μ (44) implies μ is a ifiitesimal asymptotic extremal represetative of V.Setb( V) = k ad γ (t) : [0, k] AZ () ; t [[ tμ k ]]. (45) It is clear that γ(0) = [[0]] ad γ(k) = [[μ]] ;moreover, γ([0, k]) is a geodesic segmet joiig 0 ad V i AZ(). Let p D ad U r (p) = { D : p < r}. I this sectio, we will discuss the ouiqueess of geodesic segmets betwee 0 ad V i AZ(D) whe V is a ifiitesimal osubstatial poit; that is, there exists a poit p Dwith b p ( V) < b( V). We eed the followig importat lemma. Lemma 2. Let V AZ(D) ad p D. The,foray give ε>0, there exists a ifiitesimal asymptotic extremal represetative μ of V such that b p (μ)< b p ( V) + ε. (46) Proof. Suppose p is a ifiitesimal substatial boudary poit for V;thatis,b p ( V) = b( V). It is kow i [6] that there exists a ifiitesimal asymptotic extremal represetative μ L (D) such that b( V) = b (μ). Wecocludethat(46)holds sice b p (μ) b (μ) =b( V) =b p ( V). (47) Otherwise, suppose b p ( V) < b( V).Forayε>0,without loss of geerality, we assume that ε<b( V) b p ( V). Bythe defiitio of the ifiitesimal local boudary dilatatio, there existsa Beltramidifferetialμ represetig V such that b p (μ)< b p ( V) + ε 2 <b( V). (48) Moreover, by the defiitio of b p (μ), there exists r 0 >0such that μ U r0 (p) <b p (μ)+ ε 2. (49) Let ] be a ifiitesimal extremal ad asymptotic extremal represetative of V;thatis, ] =b( V), ad let { } Q d () be a asymptotic Hamilto sequece of ].ByLemma1(r<r 0 sufficietly small), we have So dx dy = 0. (50) U r (p) ] dx dy U r (p) ] =0, U r (p) dx dy (51)

5 Joural of Fuctio Spaces 5 which meas ] dx dy = 0. (52) U r (p) There exists a boudary poit p satisfyig D outside U r (p) b p ( V) = b ( V), (53) so we have b p (V) =b (V).Letχ be the characteristic fuctio ad ] = ]χ D\Ur (p).the Sice b ( ]) =b (]) =b( V). (54) ] dx dy = ] dx dy + ] dx dy, (55) D D U r (p) we get ] dx dy = ] D dx dy, (56) D due to (52). It follows from (18) ad (54) that ] dx dy = D ] dx dy =b( V) D =b ( ]). (57) So ] is a ifiitesimal asymptotically extremal Beltrami differetial i its equivalece class [[ ]]] ad b([[ ]]] )=b( V). Defie μ () = { ] (), D \U r (p); { μ { (), U r (p). (58) It is ot hard to verify that μ() is a ifiitesimal asymptotically extremal represetative of V, ad by (48) ad (49), b p ( μ) < b p( V) + ε.thiscompletestheproofoflemma2. Theorem 3. For every V AZ(D), if V is a osubstatial poit, that is, b p ( V) < b( V) for some poit p D, the there exist ifiitely may geodesic segmets coectig V ad the basepoit i AZ(D). Proof. Let ε = b( V) b p ( V). By Lemma 2, there exists a ifiitesimal asymptotic extremal represetative μ of V such that b p (μ)< b p ( V) + ε. (59) Set b( V) = k, μ Ur0 = k ad δ = 1 k /k. Sice μ is ifiitesimally asymptotically extremal, b (μ) = b(v) = k. From (18), there exists a asymptotic Hamilto sequece {ψ } Q d () such that μψ =b (μ). (61) Furthermore, by Lemma 1, we have U r0 ψ =0, (62) for sufficietly small r 0. Let χ r0 be the characteristic fuctio of U r0 ad let be the special sequece degeeratig towards p costructed as above. For every 0 ρ δad t [0,k],let γ ρ (t) =[[ tμ k + (ρ/2)t (k t) χ r 0 ]], (63) where = =1 is a holomorphic quadratic differetial o D.Clearly,γ ρ (0) = [[0]] ad γ ρ (k) = [[μ]]. Now we show that γ ρ ([0, k]) is a geodesic segmet i AZ(D). Let0 t 1 t 2 k.wediscusstheifiitesimally asymptotic equivalece class [[(t 2 t 1 )( μ k + (ρ/2)(k t 1 t 2 )χ r0 )]]. (64) By Lemma 1, it is easy to calculate that D Sice (t 2 t 1 )( μ k + (ρ/2)(k t 1 t 2 )χ r ) ψ =t 2 t 1. b ((t 2 t 1 )( μ k + (ρ/2)(k t 1 t 2 )χ r0 )) =t 2 t 1, we obtai that (65) (66) (t 2 t 1 )( μ k + (ρ/2)(k t 1 t 2 )χ r0 ) (67) is ifiitesimally asymptotically extremal i its equivalece class. From (16), we have d(γ ρ (t 1 ),γ ρ (t 2 )) = b ([[(t 2 t 1 ) So there exists a positive umber r 0 ad a eighbourhood U r0 ={ D : p <r 0 } of p such that μ U r0 <b( V). (60) ( μ k + (ρ/2)(k t 1 t 2 )χ r0 )]] )=t 2 t 1, (68)

6 6 Joural of Fuctio Spaces which implies that γ ρ ([0, k]) is a geodesic segmet joiig [[0]] ad [[μ]] i AZ(D). Now we prove that, for 0 ρ 1 <ρ 2 δ, γ ρ1 ([0, k]) ad γ ρ2 ([0, k]) are two differet geodesics joiig [[0]] ad [[μ]] i AZ(D). Otherwise, suppose γ ρ1 (t) = γ ρ2 (t), t (0, k) whe ρ 1 <ρ 2.The [[ ((ρ 2 ρ 1 )/2)t (k t) χ r0 ]] Z 0 (D). (69) From (15), for the special sequece degeeratig towards p costructed as above, it yields χ r0 D =0. (70) O the other had, there exists a positive umber sequece {r } correspodig to { } with r <r 1 < <r 0 ad r =0such that χ r0 D = U r0 \U r 1 + U r 1 \Ur It follows from (38) that So + U r. U r =0, U r U r0 \U r 1 =0. U r0 \U r 1 Furthermore, U r =0, U r0 \U r 1 =0. U r 1 \U r = U r 1 \U r It follows from (41) ad (42) that + U r 1 \U r ( ). (71) (72) (73) (74) U r 1 \U r ( )=0, (75) =1. U r 1 \U r Therefore, By (71) (76), we coclude that U r 1 \U r =1. (76) χ r0 D =1. (77) This is a cotradictio with (70), which implies γ ρ1 (t) =γ ρ2 (t) for t (0, k) if ρ 1 =ρ 2. Thus, we have costructed ifiitely may geodesics γ ρ (t) (0 ρ δ)joiig[[0]] ad [[μ]] i AZ(D). The situatio o the geodesics joiig a ifiitesimal substatial poit with the basepoit is ot clear. We cojecture that there exist ifiitely may geodesics betwee a ifiitesimal substatial poit ad the basepoit i AZ(D). Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmet The research is partially supported by the Natioal Natural Sciece Foudatio of Chia (Grats os , , ad ). efereces [1] C. J. Earle, F. P. Gardier, ad N. Lakic, Asymptotic Teichmüller space. Part II: The metric structure, Cotemporary Mathematics,vol.355,pp ,2004. [2] K. Astala ad M. Zismeister, Teichmüller spaces ad BMOA, Mathematische Aale,vol.289,o.4,pp ,1991. [3] G. Cui, Itegrably asymptotic affie homeomorphisms of the circle ad Teichmüller spaces, Sciece i Chia Series A: Mathematics,vol.43,o.3,pp ,2000. [4] A. Fletcher, O asymptotic Teichmüller space, Trasactios of the America Mathematical Society, vol.362,o.5,pp , [5] H. Miyachi, O ivariat distaces o asymptotic Teichmüller spaces, Proceedigs of the America Mathematical Society, vol. 134, o. 7, pp , [6] F. P. Gardier ad N. Lakic, Quasicoformal Teichmüller Theory, America Mathematical Society, Providece, I, USA, [7] O. Lehto, Uivalet Fuctios ad Teichmüller Spaces,Spriger, New York, NY, USA, [8] S. Nag, The Complex Aalytic Theory of Teichmüller Spaces, Wiley-Itersciece, [9] C.J.Earle,I.Kra,adS.L.Krushkaĺ, Holomorphic motios ad teichmüller spaces, Trasactios of the America Mathematical Society,vol.343,o.2,pp ,1994. [10] J. Fa, Ogeodesics iasymptoticteichmüller spaces, Mathematische Zeitschrift, vol. 267, o. 3-4, pp , 2011.

7 Joural of Fuctio Spaces 7 [11] Z. Li, Nouiqueess of geodesics i ifiite dimesioal Teichmüller spaces, Complex Variables, Theory ad Applicatio,vol.16,o.4,pp ,1991. [12] Z. Li, No-uiqueess of geodesics i ifiite dimesioalteichmüller spaces (II), Aales Academiæ Scietiarum Feicæ Mathematica,vol.18,pp ,1993. [13] Y. L. She, Some remarks o the geodesics i ifiitedimesioalteichmüller spaces, Acta Mathematica Siica,vol. 13,o.4,pp ,1997. [14] Y.-L. She, O Teichmüller geometry, Complex Variables, Theory ad Applicatio,vol.44,o.1,pp.73 83,2001. [15] H. Taigawa, Holomorphic families of geodesic discs i ifiite-dimesioal Teichmüller spaces, Nagoya Mathematical Joural,vol.127,pp ,1992. [16] N. Lakic, Substatial boudary poits for plae domais ad Gardier s cojecture, Aales Academiæ Scietiarum Feicæ Mathematica,vol.25,pp ,2000. [17]. Fehlma ad K.-I. Saka, O the set of substatial boudary poits for extremal quasicoformal mappigs, Complex Variables, Theory ad Applicatio, vol.6,o.2 4,pp , [18] Z. Li, A ote o geodesics i ifiite-dimesioal Teichmüller spaces, Aales Academiae Scietiarum Feicae. Series A I. Mathematica,vol.20,o.2,pp ,1995.

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