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

Size: px
Start display at page:

Download "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"

Transcription

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

ABELIAN COVERINGS, POINCARÉ EXPONENT OF CONVERGENCE AND HOLOMORPHIC DEFORMATIONS

ABELIAN COVERINGS, POINCARÉ EXPONENT OF CONVERGENCE AND HOLOMORPHIC DEFORMATIONS Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 81 86 ABELIAN COVERINGS, POINCARÉ EXPONENT OF CONVERGENCE AND HOLOMORPHIC DEFORMATIONS K. Astala and M. Zinsmeister University

More information

Norwegian University of Science and Technology N-7491 Trondheim, Norway

Norwegian University of Science and Technology N-7491 Trondheim, Norway QUASICONFORMAL GEOMETRY AND DYNAMICS BANACH CENTER PUBLICATIONS, VOLUME 48 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1999 WHAT IS A DISK? KARI HAG Norwegian University of Science and

More information

Stream lines, quasilines and holomorphic motions

Stream lines, quasilines and holomorphic motions arxiv:1407.1561v1 [math.cv] 7 Jul 014 Stream lines, quasilines and holomorphic motions Gaven J. Martin Abstract We give a new application of the theory of holomorphic motions to the study the distortion

More information

Quasi-conformal maps and Beltrami equation

Quasi-conformal maps and Beltrami equation Chapter 7 Quasi-conformal maps and Beltrami equation 7. Linear distortion Assume that f(x + iy) =u(x + iy)+iv(x + iy) be a (real) linear map from C C that is orientation preserving. Let z = x + iy and

More information

ON THE NIELSEN-THURSTON-BERS TYPE OF SOME SELF-MAPS OF RIEMANN SURFACES WITH TWO SPECIFIED POINTS

ON THE NIELSEN-THURSTON-BERS TYPE OF SOME SELF-MAPS OF RIEMANN SURFACES WITH TWO SPECIFIED POINTS Imayoshi, Y., Ito, M. and Yamamoto, H. Osaka J. Math. 40 (003), 659 685 ON THE NIELSEN-THURSTON-BERS TYPE OF SOME SELF-MAPS OF RIEMANN SURFACES WITH TWO SPECIFIED POINTS Dedicated to Professor Hiroki Sato

More information

Masahiro Yanagishita. Riemann surfaces and Discontinuous groups 2014 February 16th, 2015

Masahiro Yanagishita. Riemann surfaces and Discontinuous groups 2014 February 16th, 2015 Complex analytic structure on the p-integrable Teichmüller space Masahiro Yanagishita Department of Mathematics, Waseda University JSPS Research Fellow Riemann surfaces and Discontinuous groups 2014 February

More information

4.6 Montel's Theorem. Robert Oeckl CA NOTES 7 17/11/2009 1

4.6 Montel's Theorem. Robert Oeckl CA NOTES 7 17/11/2009 1 Robert Oeckl CA NOTES 7 17/11/2009 1 4.6 Montel's Theorem Let X be a topological space. We denote by C(X) the set of complex valued continuous functions on X. Denition 4.26. A topological space is called

More information

ON A DISTANCE DEFINED BY THE LENGTH SPECTRUM ON TEICHMÜLLER SPACE

ON A DISTANCE DEFINED BY THE LENGTH SPECTRUM ON TEICHMÜLLER SPACE Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 315 326 ON A DISTANCE DEFINED BY THE LENGTH SPECTRUM ON TEICHMÜLLER SPACE Hiroshige Shiga Tokyo Institute of Technology, Department of

More information

COMPLETELY INVARIANT JULIA SETS OF POLYNOMIAL SEMIGROUPS

COMPLETELY INVARIANT JULIA SETS OF POLYNOMIAL SEMIGROUPS Series Logo Volume 00, Number 00, Xxxx 19xx COMPLETELY INVARIANT JULIA SETS OF POLYNOMIAL SEMIGROUPS RICH STANKEWITZ Abstract. Let G be a semigroup of rational functions of degree at least two, under composition

More information

THE HYPERBOLIC METRIC OF A RECTANGLE

THE HYPERBOLIC METRIC OF A RECTANGLE Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 26, 2001, 401 407 THE HYPERBOLIC METRIC OF A RECTANGLE A. F. Beardon University of Cambridge, DPMMS, Centre for Mathematical Sciences Wilberforce

More information

SOME COUNTEREXAMPLES IN DYNAMICS OF RATIONAL SEMIGROUPS. 1. Introduction

SOME COUNTEREXAMPLES IN DYNAMICS OF RATIONAL SEMIGROUPS. 1. Introduction SOME COUNTEREXAMPLES IN DYNAMICS OF RATIONAL SEMIGROUPS RICH STANKEWITZ, TOSHIYUKI SUGAWA, AND HIROKI SUMI Abstract. We give an example of two rational functions with non-equal Julia sets that generate

More information

1 Holomorphic functions

1 Holomorphic functions Robert Oeckl CA NOTES 1 15/09/2009 1 1 Holomorphic functions 11 The complex derivative The basic objects of complex analysis are the holomorphic functions These are functions that posses a complex derivative

More information

Citation 数理解析研究所講究録 (2008), 1605:

Citation 数理解析研究所講究録 (2008), 1605: Title Drawing the complex projective stru tori (Geometry related to the theor Author(s) Komori, Yohei Citation 数理解析研究所講究録 (2008), 1605: 81-89 Issue Date 2008-06 URL http://hdl.handle.net/2433/139947 Right

More information

INTRODUCTION TO REAL ANALYTIC GEOMETRY

INTRODUCTION TO REAL ANALYTIC GEOMETRY INTRODUCTION TO REAL ANALYTIC GEOMETRY KRZYSZTOF KURDYKA 1. Analytic functions in several variables 1.1. Summable families. Let (E, ) be a normed space over the field R or C, dim E

More information

Abstract. We derive a sharp bound for the modulus of the Schwarzian derivative of concave univalent functions with opening angle at infinity

Abstract. We derive a sharp bound for the modulus of the Schwarzian derivative of concave univalent functions with opening angle at infinity A SHARP BOUND FOR THE SCHWARZIAN DERIVATIVE OF CONCAVE FUNCTIONS BAPPADITYA BHOWMIK AND KARL-JOACHIM WIRTHS Abstract. We derive a sharp bound for the modulus of the Schwarzian derivative of concave univalent

More information

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X. Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2

More information

DAVID MAPS AND HAUSDORFF DIMENSION

DAVID MAPS AND HAUSDORFF DIMENSION Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 9, 004, 38 DAVID MAPS AND HAUSDORFF DIMENSION Saeed Zakeri Stony Brook University, Institute for Mathematical Sciences Stony Brook, NY 794-365,

More information

Notes on projective structures and Kleinian groups

Notes on projective structures and Kleinian groups Notes on projective structures and Kleinian groups Katsuhiko Matsuzaki and John A. Velling 1 Introduction Throughout this paper, C will denote the complex plane, Ĉ = C { } the number sphere, and D = {z

More information

Quasiconformal Maps and Circle Packings

Quasiconformal Maps and Circle Packings Quasiconformal Maps and Circle Packings Brett Leroux June 11, 2018 1 Introduction Recall the statement of the Riemann mapping theorem: Theorem 1 (Riemann Mapping). If R is a simply connected region in

More information

An Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010

An Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010 An Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010 John P. D Angelo, Univ. of Illinois, Urbana IL 61801.

More information

Bounded and integrable quadratic differentials: hyperbolic and extremal lengths on Riemann surfaces

Bounded and integrable quadratic differentials: hyperbolic and extremal lengths on Riemann surfaces Geometric Complex Analysis edited by Junjiro Noguchi et al. World Scientific, Singapore, 1995 pp.443 450 Bounded and integrable quadratic differentials: hyperbolic and extremal lengths on Riemann surfaces

More information

Bloch radius, normal families and quasiregular mappings

Bloch radius, normal families and quasiregular mappings Bloch radius, normal families and quasiregular mappings Alexandre Eremenko Abstract Bloch s Theorem is extended to K-quasiregular maps R n S n, where S n is the standard n-dimensional sphere. An example

More information

Coarse Geometry. Phanuel Mariano. Fall S.i.g.m.a. Seminar. Why Coarse Geometry? Coarse Invariants A Coarse Equivalence to R 1

Coarse Geometry. Phanuel Mariano. Fall S.i.g.m.a. Seminar. Why Coarse Geometry? Coarse Invariants A Coarse Equivalence to R 1 Coarse Geometry 1 1 University of Connecticut Fall 2014 - S.i.g.m.a. Seminar Outline 1 Motivation 2 3 The partition space P ([a, b]). Preliminaries Main Result 4 Outline Basic Problem 1 Motivation 2 3

More information

Conformal Mappings. Chapter Schwarz Lemma

Conformal Mappings. Chapter Schwarz Lemma Chapter 5 Conformal Mappings In this chapter we study analytic isomorphisms. An analytic isomorphism is also called a conformal map. We say that f is an analytic isomorphism of U with V if f is an analytic

More information

COMPOSITION SEMIGROUPS ON BMOA AND H AUSTIN ANDERSON, MIRJANA JOVOVIC, AND WAYNE SMITH

COMPOSITION SEMIGROUPS ON BMOA AND H AUSTIN ANDERSON, MIRJANA JOVOVIC, AND WAYNE SMITH COMPOSITION SEMIGROUPS ON BMOA AND H AUSTIN ANDERSON, MIRJANA JOVOVIC, AND WAYNE SMITH Abstract. We study [ϕ t, X], the maximal space of strong continuity for a semigroup of composition operators induced

More information

Garrett: `Bernstein's analytic continuation of complex powers' 2 Let f be a polynomial in x 1 ; : : : ; x n with real coecients. For complex s, let f

Garrett: `Bernstein's analytic continuation of complex powers' 2 Let f be a polynomial in x 1 ; : : : ; x n with real coecients. For complex s, let f 1 Bernstein's analytic continuation of complex powers c1995, Paul Garrett, garrettmath.umn.edu version January 27, 1998 Analytic continuation of distributions Statement of the theorems on analytic continuation

More information

Mapping problems and harmonic univalent mappings

Mapping problems and harmonic univalent mappings Mapping problems and harmonic univalent mappings Antti Rasila Helsinki University of Technology antti.rasila@tkk.fi (Mainly based on P. Duren s book Harmonic mappings in the plane) Helsinki Analysis Seminar,

More information

Ioannis D. Platis (Received February 2003)

Ioannis D. Platis (Received February 2003) NEW ZEALAND JOURNAL OF MATHEMATICS Volume 35 2006, 85 107 REMARKS ON THE HYPERBOLIC GEOMETRY OF PRODUCT TEICHMÜLLER SPACES Ioannis D. Platis Received February 2003 Abstract. Let e T be a cross product

More information

(1.) For any subset P S we denote by L(P ) the abelian group of integral relations between elements of P, i.e. L(P ) := ker Z P! span Z P S S : For ea

(1.) For any subset P S we denote by L(P ) the abelian group of integral relations between elements of P, i.e. L(P ) := ker Z P! span Z P S S : For ea Torsion of dierentials on toric varieties Klaus Altmann Institut fur reine Mathematik, Humboldt-Universitat zu Berlin Ziegelstr. 13a, D-10099 Berlin, Germany. E-mail: altmann@mathematik.hu-berlin.de Abstract

More information

ON COMPACTNESS OF THE DIFFERENCE OF COMPOSITION OPERATORS. C φ 2 e = lim sup w 1

ON COMPACTNESS OF THE DIFFERENCE OF COMPOSITION OPERATORS. C φ 2 e = lim sup w 1 ON COMPACTNESS OF THE DIFFERENCE OF COMPOSITION OPERATORS PEKKA NIEMINEN AND EERO SAKSMAN Abstract. We give a negative answer to a conjecture of J. E. Shapiro concerning compactness of the dierence of

More information

Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative

Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative This chapter is another review of standard material in complex analysis. See for instance

More information

11 COMPLEX ANALYSIS IN C. 1.1 Holomorphic Functions

11 COMPLEX ANALYSIS IN C. 1.1 Holomorphic Functions 11 COMPLEX ANALYSIS IN C 1.1 Holomorphic Functions A domain Ω in the complex plane C is a connected, open subset of C. Let z o Ω and f a map f : Ω C. We say that f is real differentiable at z o if there

More information

SCHWARZIAN DERIVATIVES OF CONVEX MAPPINGS

SCHWARZIAN DERIVATIVES OF CONVEX MAPPINGS Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 36, 2011, 449 460 SCHWARZIAN DERIVATIVES OF CONVEX MAPPINGS Martin Chuaqui, Peter Duren and Brad Osgood P. Universidad Católica de Chile, Facultad

More information

David E. Barrett and Jeffrey Diller University of Michigan Indiana University

David E. Barrett and Jeffrey Diller University of Michigan Indiana University A NEW CONSTRUCTION OF RIEMANN SURFACES WITH CORONA David E. Barrett and Jeffrey Diller University of Michigan Indiana University 1. Introduction An open Riemann surface X is said to satisfy the corona

More information

ON HARMONIC FUNCTIONS ON SURFACES WITH POSITIVE GAUSS CURVATURE AND THE SCHWARZ LEMMA

ON HARMONIC FUNCTIONS ON SURFACES WITH POSITIVE GAUSS CURVATURE AND THE SCHWARZ LEMMA ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 44, Number 5, 24 ON HARMONIC FUNCTIONS ON SURFACES WITH POSITIVE GAUSS CURVATURE AND THE SCHWARZ LEMMA DAVID KALAJ ABSTRACT. We prove some versions of the Schwarz

More information

1. If 1, ω, ω 2, -----, ω 9 are the 10 th roots of unity, then (1 + ω) (1 + ω 2 ) (1 + ω 9 ) is A) 1 B) 1 C) 10 D) 0

1. If 1, ω, ω 2, -----, ω 9 are the 10 th roots of unity, then (1 + ω) (1 + ω 2 ) (1 + ω 9 ) is A) 1 B) 1 C) 10 D) 0 4 INUTES. If, ω, ω, -----, ω 9 are the th roots of unity, then ( + ω) ( + ω ) ----- ( + ω 9 ) is B) D) 5. i If - i = a + ib, then a =, b = B) a =, b = a =, b = D) a =, b= 3. Find the integral values for

More information

neighborhood of the point or do not. The grand orbit of any component of the Fatou set contains one of the following domains; an attractive basin, a s

neighborhood of the point or do not. The grand orbit of any component of the Fatou set contains one of the following domains; an attractive basin, a s On Teichmuller spaces of complex dynamics by entire functions Tatsunori Harada and Masahiko Taniguchi Abstract In this paper, we give avery brief exposition of the general Teichmuller theory for complex

More information

Igor E. Pritsker. plane, we consider a problem of the uniform approximation on E by

Igor E. Pritsker. plane, we consider a problem of the uniform approximation on E by Weighted Approximation on Compact Sets Igor E. Pritsker Abstract. For a compact set E with connected complement in the complex plane, we consider a problem of the uniform approximation on E by the weighted

More information

Tools from Lebesgue integration

Tools from Lebesgue integration Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given

More information

Quasiconformal and Lipschitz harmonic mappings of the unit disk onto bounded convex domains

Quasiconformal and Lipschitz harmonic mappings of the unit disk onto bounded convex domains Quasiconformal and Lipschitz harmonic mappings of the unit disk onto bounded convex domains Ken-ichi Sakan (Osaka City Univ., Japan) Dariusz Partyka (The John Paul II Catholic University of Lublin, Poland)

More information

A TALE OF TWO CONFORMALLY INVARIANT METRICS

A TALE OF TWO CONFORMALLY INVARIANT METRICS A TALE OF TWO CONFORMALLY INVARIANT METRICS H. S. BEAR AND WAYNE SMITH Abstract. The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic

More information

POWER SERIES AND ANALYTIC CONTINUATION

POWER SERIES AND ANALYTIC CONTINUATION POWER SERIES AND ANALYTIC CONTINUATION 1. Analytic functions Definition 1.1. A function f : Ω C C is complex-analytic if for each z 0 Ω there exists a power series f z0 (z) := a n (z z 0 ) n which converges

More information

A Note on the Harmonic Quasiconformal Diffeomorphisms of the Unit Disc

A Note on the Harmonic Quasiconformal Diffeomorphisms of the Unit Disc Filomat 29:2 (2015), 335 341 DOI 10.2298/FIL1502335K Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Note on the Harmonic Quasiconformal

More information

ON INFINITESIMAL STREBEL POINTS

ON INFINITESIMAL STREBEL POINTS ON ININITSIMAL STRBL POINTS ALASTAIR LTCHR Abstract. In this paper, we prove that if is a Riemann surface of infinite analytic type and [µ] T is any element of Teichmüller space, then there exists µ 1

More information

ON THE MEAN CURVATURE FUNCTION FOR COMPACT SURFACES

ON THE MEAN CURVATURE FUNCTION FOR COMPACT SURFACES J. DIFFERENTIAL GEOMETRY 16 (1981) 179-183 ON THE MEAN CURVATURE FUNCTION FOR COMPACT SURFACES H. BLAINE LAWSON, JR. & RENATO DE AZEVEDO TRIBUZY Dedicated to Professor Buchin Su on his SOth birthday It

More information

RANDOM HOLOMORPHIC ITERATIONS AND DEGENERATE SUBDOMAINS OF THE UNIT DISK

RANDOM HOLOMORPHIC ITERATIONS AND DEGENERATE SUBDOMAINS OF THE UNIT DISK PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 RANDOM HOLOMORPHIC ITERATIONS AND DEGENERATE SUBDOMAINS OF THE UNIT DISK LINDA KEEN AND NIKOLA

More information

William P. Thurston. The Geometry and Topology of Three-Manifolds

William P. Thurston. The Geometry and Topology of Three-Manifolds William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 00 http://www.msri.org/publications/books/gt3m/ This is an electronic edition of the 1980 notes distributed

More information

94. E. Marshall and W. Smith The problem considered in this paper concerns the angular distribution of mass by such a measure. For ">0 we dene " = fz

94. E. Marshall and W. Smith The problem considered in this paper concerns the angular distribution of mass by such a measure. For >0 we dene  = fz Revista Matematica Iberoamericana Vol. 15, N. o 1, 1999 The angular distribution of mass by Bergman functions onald E. Marshall and Wayne Smith Abstract. Let = fz : jzj < 1g be the unit disk in the complex

More information

Free Boundary Value Problems for Analytic Functions in the Closed Unit Disk

Free Boundary Value Problems for Analytic Functions in the Closed Unit Disk Free Boundary Value Problems for Analytic Functions in the Closed Unit Disk Richard Fournier Stephan Ruscheweyh CRM-2558 January 1998 Centre de recherches de mathématiques, Université de Montréal, Montréal

More information

Accumulation constants of iterated function systems with Bloch target domains

Accumulation constants of iterated function systems with Bloch target domains Accumulation constants of iterated function systems with Bloch target domains September 29, 2005 1 Introduction Linda Keen and Nikola Lakic 1 Suppose that we are given a random sequence of holomorphic

More information

ON THE MARGULIS CONSTANT FOR KLEINIAN GROUPS, I

ON THE MARGULIS CONSTANT FOR KLEINIAN GROUPS, I Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 439 462 ON THE MARGULIS CONSTANT FOR KLEINIAN GROUPS, I F.W. Gehring and G.J. Martin University of Michigan, Department of Mathematics

More information

MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5

MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5 MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5.. The Arzela-Ascoli Theorem.. The Riemann mapping theorem Let X be a metric space, and let F be a family of continuous complex-valued functions on X. We have

More information

2 G. D. DASKALOPOULOS AND R. A. WENTWORTH general, is not true. Thus, unlike the case of divisors, there are situations where k?1 0 and W k?1 = ;. r;d

2 G. D. DASKALOPOULOS AND R. A. WENTWORTH general, is not true. Thus, unlike the case of divisors, there are situations where k?1 0 and W k?1 = ;. r;d ON THE BRILL-NOETHER PROBLEM FOR VECTOR BUNDLES GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH Abstract. On an arbitrary compact Riemann surface, necessary and sucient conditions are found for the

More information

arxiv: v1 [math.cv] 12 Jul 2007

arxiv: v1 [math.cv] 12 Jul 2007 A NOTE ON THE HAYMAN-WU THEOREM EDWARD CRANE arxiv:0707.1772v1 [math.cv] 12 Jul 2007 Abstract. The Hayman-Wu theorem states that the preimage of a line or circle L under a conformal mapping from the unit

More information

THE RIEMANN MAPPING THEOREM

THE RIEMANN MAPPING THEOREM THE RIEMANN MAPPING THEOREM ALEXANDER WAUGH Abstract. This paper aims to provide all necessary details to give the standard modern proof of the Riemann Mapping Theorem utilizing normal families of functions.

More information

WELL-POSEDNESS OF A RIEMANN HILBERT

WELL-POSEDNESS OF A RIEMANN HILBERT Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 42, 207, 4 47 WELL-POSEDNESS OF A RIEMANN HILBERT PROBLEM ON d-regular QUASIDISKS Eric Schippers and Wolfgang Staubach University of Manitoba, Department

More information

arxiv: v1 [math.cv] 17 Nov 2016

arxiv: v1 [math.cv] 17 Nov 2016 arxiv:1611.05667v1 [math.cv] 17 Nov 2016 CRITERIA FOR BOUNDED VALENCE OF HARMONIC MAPPINGS JUHA-MATTI HUUSKO AND MARÍA J. MARTÍN Abstract. In 1984, Gehring and Pommerenke proved that if the Schwarzian

More information

Fuchsian groups. 2.1 Definitions and discreteness

Fuchsian groups. 2.1 Definitions and discreteness 2 Fuchsian groups In the previous chapter we introduced and studied the elements of Mob(H), which are the real Moebius transformations. In this chapter we focus the attention of special subgroups of this

More information

Qualifying Exam Complex Analysis (Math 530) January 2019

Qualifying Exam Complex Analysis (Math 530) January 2019 Qualifying Exam Complex Analysis (Math 53) January 219 1. Let D be a domain. A function f : D C is antiholomorphic if for every z D the limit f(z + h) f(z) lim h h exists. Write f(z) = f(x + iy) = u(x,

More information

Möbius transformations Möbius transformations are simply the degree one rational maps of C: cz + d : C C. ad bc 0. a b. A = c d

Möbius transformations Möbius transformations are simply the degree one rational maps of C: cz + d : C C. ad bc 0. a b. A = c d Möbius transformations Möbius transformations are simply the degree one rational maps of C: where and Then σ A : z az + b cz + d : C C ad bc 0 ( ) a b A = c d A σ A : GL(2C) {Mobius transformations } is

More information

UNIFORMLY PERFECT ANALYTIC AND CONFORMAL ATTRACTOR SETS. 1. Introduction and results

UNIFORMLY PERFECT ANALYTIC AND CONFORMAL ATTRACTOR SETS. 1. Introduction and results UNIFORMLY PERFECT ANALYTIC AND CONFORMAL ATTRACTOR SETS RICH STANKEWITZ Abstract. Conditions are given which imply that analytic iterated function systems (IFS s) in the complex plane C have uniformly

More information

MATH 566 LECTURE NOTES 6: NORMAL FAMILIES AND THE THEOREMS OF PICARD

MATH 566 LECTURE NOTES 6: NORMAL FAMILIES AND THE THEOREMS OF PICARD MATH 566 LECTURE NOTES 6: NORMAL FAMILIES AND THE THEOREMS OF PICARD TSOGTGEREL GANTUMUR 1. Introduction Suppose that we want to solve the equation f(z) = β where f is a nonconstant entire function and

More information

COMPACT DIFFERENCE OF WEIGHTED COMPOSITION OPERATORS ON N p -SPACES IN THE BALL

COMPACT DIFFERENCE OF WEIGHTED COMPOSITION OPERATORS ON N p -SPACES IN THE BALL COMPACT DIFFERENCE OF WEIGHTED COMPOSITION OPERATORS ON N p -SPACES IN THE BALL HU BINGYANG and LE HAI KHOI Communicated by Mihai Putinar We obtain necessary and sucient conditions for the compactness

More information

ondary 31C05 Key words and phrases: Planar harmonic mappings, Quasiconformal mappings, Planar domains

ondary 31C05 Key words and phrases: Planar harmonic mappings, Quasiconformal mappings, Planar domains Novi Sad J. Math. Vol. 38, No. 3, 2008, 147-156 QUASICONFORMAL AND HARMONIC MAPPINGS BETWEEN SMOOTH JORDAN DOMAINS David Kalaj 1, Miodrag Mateljević 2 Abstract. We present some recent results on the topic

More information

Doc. Math. J. DMV 321 Chern Classes of Fibered Products of Surfaces Mina Teicher Received: March 16, 1998 Revised: September 18, 1998 Communicated by

Doc. Math. J. DMV 321 Chern Classes of Fibered Products of Surfaces Mina Teicher Received: March 16, 1998 Revised: September 18, 1998 Communicated by Doc. Math. J. DMV 31 Chern Classes of Fibered Products of Surfaces Mina Teicher Received: March 16, 1998 Revised: September 18, 1998 Communicated by Thomas Peternell Abstract. In this paper we introduce

More information

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite

More information

BEST APPROXIMATIONS AND ORTHOGONALITIES IN 2k-INNER PRODUCT SPACES. Seong Sik Kim* and Mircea Crâşmăreanu. 1. Introduction

BEST APPROXIMATIONS AND ORTHOGONALITIES IN 2k-INNER PRODUCT SPACES. Seong Sik Kim* and Mircea Crâşmăreanu. 1. Introduction Bull Korean Math Soc 43 (2006), No 2, pp 377 387 BEST APPROXIMATIONS AND ORTHOGONALITIES IN -INNER PRODUCT SPACES Seong Sik Kim* and Mircea Crâşmăreanu Abstract In this paper, some characterizations of

More information

DECOMPOSING DIFFEOMORPHISMS OF THE SPHERE. inf{d Y (f(x), f(y)) : d X (x, y) r} H

DECOMPOSING DIFFEOMORPHISMS OF THE SPHERE. inf{d Y (f(x), f(y)) : d X (x, y) r} H DECOMPOSING DIFFEOMORPHISMS OF THE SPHERE ALASTAIR FLETCHER, VLADIMIR MARKOVIC 1. Introduction 1.1. Background. A bi-lipschitz homeomorphism f : X Y between metric spaces is a mapping f such that f and

More information

arxiv:math/ v1 [math.fa] 26 Oct 1993

arxiv:math/ v1 [math.fa] 26 Oct 1993 arxiv:math/9310217v1 [math.fa] 26 Oct 1993 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M.I.Ostrovskii Abstract. It is proved that there exist complemented subspaces of countable topological

More information

ON HEINZ S INEQUALITY

ON HEINZ S INEQUALITY ON HEINZ S INEQUALITY DARIUSZ PARTYKA AND KEN-ICHI SAKAN In memory of Professor Zygmunt Charzyński Abstract. In 1958 E. Heinz obtained a lower bound for xf + yf, where F isaone-to-oneharmonicmappingoftheunitdiscontoitselfkeeping

More information

The bumping set and the characteristic submanifold

The bumping set and the characteristic submanifold The bumping set and the characteristic submanifold Abstract We show here that the Nielsen core of the bumping set of the domain of discontinuity of a Kleinian group Γ is the boundary for the characteristic

More information

Riemann-Stieltjes Operators between Weighted Bloch and Weighted Bergman Spaces

Riemann-Stieltjes Operators between Weighted Bloch and Weighted Bergman Spaces Int. J. Contemp. Math. Sci., Vol. 2, 2007, no. 16, 759-772 Riemann-Stieltjes Operators between Weighted Bloch and Weighted Bergman Spaces Ajay K. Sharma 1 School of Applied Physics and Mathematics Shri

More information

Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative

Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative This chapter is another review of standard material in complex analysis. See for instance

More information

Let X be a topological space. We want it to look locally like C. So we make the following definition.

Let X be a topological space. We want it to look locally like C. So we make the following definition. February 17, 2010 1 Riemann surfaces 1.1 Definitions and examples Let X be a topological space. We want it to look locally like C. So we make the following definition. Definition 1. A complex chart on

More information

ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES

ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 7, July 1996 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M. I. OSTROVSKII (Communicated by Dale Alspach) Abstract.

More information

FENCHEL NIELSEN COORDINATES FOR ASYMPTOTICALLY CONFORMAL DEFORMATIONS

FENCHEL NIELSEN COORDINATES FOR ASYMPTOTICALLY CONFORMAL DEFORMATIONS Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 41, 2016, 167 176 FENCHEL NIELSEN COORDINATES FOR ASYMPTOTICALLY CONFORMAL DEFORMATIONS Dragomir Šarić Queens College of CUNY, Department of Mathematics

More information

arxiv: v1 [math.ca] 1 Jul 2013

arxiv: v1 [math.ca] 1 Jul 2013 Carleson measures on planar sets Zhijian Qiu Department of mathematics, Southwestern University of Finance and Economics arxiv:1307.0424v1 [math.ca] 1 Jul 2013 Abstract In this paper, we investigate what

More information

QUASICONFORMAL EXTENSION OF HARMONIC MAPPINGS IN THE PLANE

QUASICONFORMAL EXTENSION OF HARMONIC MAPPINGS IN THE PLANE Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 38, 2013, 617 630 QUASICONFORMAL EXTENSION OF HARMONIC MAPPINGS IN THE PLANE Rodrigo Hernández and María J Martín Universidad Adolfo Ibáñez, Facultad

More information

SOLUTION TO RUBEL S QUESTION ABOUT DIFFERENTIALLY ALGEBRAIC DEPENDENCE ON INITIAL VALUES GUY KATRIEL PREPRINT NO /4

SOLUTION TO RUBEL S QUESTION ABOUT DIFFERENTIALLY ALGEBRAIC DEPENDENCE ON INITIAL VALUES GUY KATRIEL PREPRINT NO /4 SOLUTION TO RUBEL S QUESTION ABOUT DIFFERENTIALLY ALGEBRAIC DEPENDENCE ON INITIAL VALUES GUY KATRIEL PREPRINT NO. 2 2003/4 1 SOLUTION TO RUBEL S QUESTION ABOUT DIFFERENTIALLY ALGEBRAIC DEPENDENCE ON INITIAL

More information

PROBLEMS. (b) (Polarization Identity) Show that in any inner product space

PROBLEMS. (b) (Polarization Identity) Show that in any inner product space 1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization

More information

A FUCHSIAN GROUP PROOF OF THE HYPERELLIPTICITY OF RIEMANN SURFACES OF GENUS 2

A FUCHSIAN GROUP PROOF OF THE HYPERELLIPTICITY OF RIEMANN SURFACES OF GENUS 2 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 69 74 A FUCHSIAN GROUP PROOF OF THE HYPERELLIPTICITY OF RIEMANN SURFACES OF GENUS 2 Yolanda Fuertes and Gabino González-Diez Universidad

More information

(03) So when the order of { divides no N j at all then Z { (s) is holomorphic on C Now the N j are not intrinsically associated to f 1 f0g; but the or

(03) So when the order of { divides no N j at all then Z { (s) is holomorphic on C Now the N j are not intrinsically associated to f 1 f0g; but the or HOLOMORPHY OF LOCAL ZETA FUNCTIONS FOR CURVES Willem Veys Introduction (01) Let K be a nite extension of the eld Q p of p{adic numbers, R the valuation ring of K, P the maximal ideal of R, and K = R=P

More information

Geometric Complex Analysis. Davoud Cheraghi Imperial College London

Geometric Complex Analysis. Davoud Cheraghi Imperial College London Geometric Complex Analysis Davoud Cheraghi Imperial College London May 9, 2017 Introduction The subject of complex variables appears in many areas of mathematics as it has been truly the ancestor of many

More information

A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS. Masaru Nagisa. Received May 19, 2014 ; revised April 10, (Ax, x) 0 for all x C n.

A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS. Masaru Nagisa. Received May 19, 2014 ; revised April 10, (Ax, x) 0 for all x C n. Scientiae Mathematicae Japonicae Online, e-014, 145 15 145 A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS Masaru Nagisa Received May 19, 014 ; revised April 10, 014 Abstract. Let f be oeprator monotone

More information

Peak Point Theorems for Uniform Algebras on Smooth Manifolds

Peak Point Theorems for Uniform Algebras on Smooth Manifolds Peak Point Theorems for Uniform Algebras on Smooth Manifolds John T. Anderson and Alexander J. Izzo Abstract: It was once conjectured that if A is a uniform algebra on its maximal ideal space X, and if

More information

Math 147, Homework 1 Solutions Due: April 10, 2012

Math 147, Homework 1 Solutions Due: April 10, 2012 1. For what values of a is the set: Math 147, Homework 1 Solutions Due: April 10, 2012 M a = { (x, y, z) : x 2 + y 2 z 2 = a } a smooth manifold? Give explicit parametrizations for open sets covering M

More information

MARIA GIRARDI Fact 1.1. For a bounded linear operator T from L 1 into X, the following statements are equivalent. (1) T is Dunford-Pettis. () T maps w

MARIA GIRARDI Fact 1.1. For a bounded linear operator T from L 1 into X, the following statements are equivalent. (1) T is Dunford-Pettis. () T maps w DENTABILITY, TREES, AND DUNFORD-PETTIS OPERATORS ON L 1 Maria Girardi University of Illinois at Urbana-Champaign Pacic J. Math. 148 (1991) 59{79 Abstract. If all bounded linear operators from L1 into a

More information

NON-DIVERGENT INFINITELY DISCRETE TEICHMÜLLER MODULAR TRANSFORMATION

NON-DIVERGENT INFINITELY DISCRETE TEICHMÜLLER MODULAR TRANSFORMATION NON-DIVERGENT INFINITELY DISCRETE TEICHMÜLLER MODULAR TRANSFORMATION EGE FUJIKAWA AND KATSUHIKO MATSUZAKI Abstract. We consider a classification of Teichmüller modular transformations of an analytically

More information

TEICHMÜLLER SPACE MATTHEW WOOLF

TEICHMÜLLER SPACE MATTHEW WOOLF TEICHMÜLLER SPACE MATTHEW WOOLF Abstract. It is a well-known fact that every Riemann surface with negative Euler characteristic admits a hyperbolic metric. But this metric is by no means unique indeed,

More information

Complex Analysis Problems

Complex Analysis Problems Complex Analysis Problems transcribed from the originals by William J. DeMeo October 2, 2008 Contents 99 November 2 2 2 200 November 26 4 3 2006 November 3 6 4 2007 April 6 7 5 2007 November 6 8 99 NOVEMBER

More information

Function Spaces - selected open problems

Function Spaces - selected open problems Contemporary Mathematics Function Spaces - selected open problems Krzysztof Jarosz Abstract. We discuss briefly selected open problems concerning various function spaces. 1. Introduction We discuss several

More information

On the Length of Lemniscates

On the Length of Lemniscates On the Length of Lemniscates Alexandre Eremenko & Walter Hayman For a monic polynomial p of degree d, we write E(p) := {z : p(z) =1}. A conjecture of Erdős, Herzog and Piranian [4], repeated by Erdős in

More information

Spectral Theory of Orthogonal Polynomials

Spectral Theory of Orthogonal Polynomials Spectral Theory of Orthogonal Polynomials Barry Simon IBM Professor of Mathematics and Theoretical Physics California Institute of Technology Pasadena, CA, U.S.A. Lecture 9: s and Finite Gaps, I Spectral

More information

Convex Analysis and Economic Theory AY Elementary properties of convex functions

Convex Analysis and Economic Theory AY Elementary properties of convex functions Division of the Humanities and Social Sciences Ec 181 KC Border Convex Analysis and Economic Theory AY 2018 2019 Topic 6: Convex functions I 6.1 Elementary properties of convex functions We may occasionally

More information

QUASINORMAL FAMILIES AND PERIODIC POINTS

QUASINORMAL FAMILIES AND PERIODIC POINTS QUASINORMAL FAMILIES AND PERIODIC POINTS WALTER BERGWEILER Dedicated to Larry Zalcman on his 60th Birthday Abstract. Let n 2 be an integer and K > 1. By f n we denote the n-th iterate of a function f.

More information

THE SLICE DETERMINED BY MODULI EQUATION xy=2z IN THE DEFORMATION SPACE OF ONCE PUNCTURED TORI

THE SLICE DETERMINED BY MODULI EQUATION xy=2z IN THE DEFORMATION SPACE OF ONCE PUNCTURED TORI Sasaki, T. Osaka J. Math. 33 (1996), 475-484 THE SLICE DETERMINED BY MODULI EQUATION xy=2z IN THE DEFORMATION SPACE OF ONCE PUNCTURED TORI TAKEHIKO SASAKI (Received February 9, 1995) 1. Introduction and

More information

Geometric measure on quasi-fuchsian groups via singular traces. Part I.

Geometric measure on quasi-fuchsian groups via singular traces. Part I. Geometric measure on quasi-fuchsian groups via singular traces. Part I. Dmitriy Zanin (in collaboration with A. Connes and F. Sukochev) University of New South Wales September 19, 2018 Dmitriy Zanin Geometric

More information

ENTRY POTENTIAL THEORY. [ENTRY POTENTIAL THEORY] Authors: Oliver Knill: jan 2003 Literature: T. Ransford, Potential theory in the complex plane.

ENTRY POTENTIAL THEORY. [ENTRY POTENTIAL THEORY] Authors: Oliver Knill: jan 2003 Literature: T. Ransford, Potential theory in the complex plane. ENTRY POTENTIAL THEORY [ENTRY POTENTIAL THEORY] Authors: Oliver Knill: jan 2003 Literature: T. Ransford, Potential theory in the complex plane. Analytic [Analytic] Let D C be an open set. A continuous

More information

BLOCH FUNCTIONS ON THE UNIT BALL OF AN INFINITE DIMENSIONAL HILBERT SPACE

BLOCH FUNCTIONS ON THE UNIT BALL OF AN INFINITE DIMENSIONAL HILBERT SPACE BLOCH FUNCTIONS ON THE UNIT BALL OF AN INFINITE DIMENSIONAL HILBERT SPACE OSCAR BLASCO, PABLO GALINDO, AND ALEJANDRO MIRALLES Abstract. The Bloch space has been studied on the open unit disk of C and some

More information

Rigidity of Teichmüller curves

Rigidity of Teichmüller curves Rigidity of Teichmüller curves Curtis T. McMullen 11 September, 2008 Let f : V M g be a holomorphic map from a Riemann surface of finite hyperbolic volume to the moduli space of compact Riemann surfaces

More information