2011 • 156 Pages • 3.76 MB • English

Posted April 14, 2020 • Uploaded
by bergnaum.catalina

PREVIEW PDF

Page 1

MOBIUS T~SFO~~TIONS IN SEVERAL DIMENSIONS ~-====~=~===~=====================~~~==~====;= . Lars V. ~hlfors

Page 2

Copyright 1981 by Lars V.. Ahlfors

Page 3

INTRODUCTION ======~===== These are day to day notes from a course that I gave as Visiting Ordway Professor at the University of Minnesota in the fall 1980. I had used some of the material earlier for a similar course at the university of Michigan and for a month each at the University of Vienna and the University of Graz. The present notes are a little more detailed, but still not in a definitive ferm. My aim was to give a rather elementary course which would acquaint the hearers with the geometric and analytic properties of Mobius transformations in n real dimensions which could serve as an introduction to discrete groups of non-euclidean motions in hyper- bolic space, especially from the viewpoint of solutions of the hyperbolically invariant Laplace equation as well as quasiconfor:mal deformations. I wish to thank the University of Minnesota and the Chairman of the Mathematics Department for having invited me, and above all my friend and former student Albert Marden for having initiated my visit. I also thank the many faculty rrembers and students who had the patience to sit in at roy lectures to the very end. I am also indebted to Dr. Helmut Maier who helped me edit the notes and supervised the typing. The typist, unknown to me, has also my sympathy. Lars V. Ahlfors

Page 4

Page 5

'TABLE OF CONTENTS ===~=~=========== I. The classical case p.l II. The general case p.13 III. Hyperbolic geometry p.31 IV. Elements of differential geometry p.41 v. Hyperharrnonic functions p.57 VI. The geodesic flow p.72 VII. Discrete subgroups p.79 VIII. Quasiconformal deformations p.101

Page 6

Page 7

I. The classical case. 1.1. Everybod¥ is familiar with the fra.ctional linear transformations az +b (1) y(z):= cz + d where a, b, c, dEC and ad - be f O. They act on the extended complex plane C = C U (CO)} which we identify with the 1 - dimensional proj eetiv e space P1 ( C) • In terms of homogeneous coordinates w = 'V(z) can be expressed through the matrix equation (2 ) This has the advantage that the Mldbius group of all y can be represented as a matrix group either by means of the ~eneral linear group G~(C) or by the unimodular or sRecial linear group S~ (C). More precisely the Mobius group is isomorphic to where C* is the multiplicative group of nonzero complex numbers and to 1 0 where r=(Ol) • In spite of this identification we shall denote the Mobius group by ~ ( C) rather than PS~ (C) for the simple reason that the identification does not carryover to higher dimensions. We shall of'ten sittplify y( z ) to yz and we think 0'£ Y as a matrix normalized by ad - bc = 1. We should be aware, of course, that y and -y

Page 8

2 represent the same M"obius t=:-a.n.sformation. It is useful to memorize the formula for the inverse in S~ : (3) 1.2. There is a natural topology on G~ (C) and hence also on S~ (C) and }~ (C). G~ ( C) is connected for the space of all complex 2 X 2 matrices has eight real dimensions and the space of singular matrices has only six. The mapping ad~ be ) of G~(C) on s~(C) shows that S~(C) is likewise connected. We can consider S~{C) as a double covering of ~~(C) • The subgroups with real coefficients, are denoted. by G~(JR) , S~(:R) and ~(:R) • There is also a subgroup Gr;C1R) with positive determinant. + G~ (:Ia) and S ~ (JR ) are connected, but G~ (JR ) is not for one cannot pass continuously from a ma~rix with positive determinant to one with negative determinant. Any YEM2(JR) maps the real axis, the upper half-plane, and the lower half-plane on themselves. This is obvious from ... z-z (4) yz - yz = 2 jcz+d·1 1.3. From (1) one computes (5) y( z) - y( z ') :=

Page 9

3 and in the limit for zt-+z ad. - be (6} y'(z) = 2 (Cz + d) In particular, z ~ yz is copf'Orr.lal. We rewrite (5) as 1/2 1/2 (7) y(z) - Y(ZI) = Y'(z) l' Y'(ZI) I' (z- z') where it is understood that := Jad- be cz+d with a f:xed choice of the square ~oot. 1 . 4. The cross-ratio of' four points z, z I , s, S' in that order is d,e:fined by zt - 6 (8) z' - (; I when tbis makes sense, i.e. When at most two points coincide. B.lf use of (7) it follow's that (z , z' , " ,I) :for the ·derivatives cancel against each other.. In other words, the cross- ratio is invariant under all Mobius tr&nsi'Qrmations .. The cross-ratio is real if and. only if the four points lie on a circle or strai~ht lines. Note that our definition makes (z , 1 , 0 ~ (0) = z

Page 10

4 1.5. From (6) one obtains y"(z) 2c (10) yl (z) cz+d and hence (11) ~:it: ~ = if c i O. It follows that (12) and for l!'" Z (13 ) D ~: t: ~ = - ~ • More explicit ,ly, this is equivalent to the vanishing of the Schwarzian derivative: III 3 It 2 II 1." 2 (14) :L _ _ (.Y.:.) = (Y...)' __ (.Y:.) ::: 0 y' 2 "VI y' 2 yl There is obviously a close link between the cross-ratio and the Schwarzian (15 ) for arbitrary meramorphic f. For those who like computing, I recommend proving the formula (16) (f(z + ta) ,f(z + tb) , f{z + tc) , fez + td)):;: 2 (a",b,c,d)(l+ 6t ' (a-b) (c-d)S:r(Z» + o(t3) .

Möbius transformations in several dimensions

2004 • 156 Pages • 3.11 MB

Random discrete groups of Möbius transformations

2017 • 101 Pages • 2.52 MB

Mastering the Discrete Fourier Transform in One, Two or Several Dimensions: Pitfalls and Artifacts

2013 • 388 Pages • 35.21 MB

Composition sequences and semigroups of Möbius transformations

2017 • 171 Pages • 5.03 MB

Several Short Sentences About Writing

2012 • 183 Pages • 576.67 KB

State Transformations in OECD Countries: Dimensions, Driving Forces and Trajectories

2015 • 319 Pages • 2.01 MB

Proceedings symposium on value distribution theory in several complex variables

1992 • 178 Pages • 5.48 MB

Proceedings Symposium on Value Distribution Theory in Several Complex Variables

1992 • 178 Pages • 11.52 MB

The Several Lives of Chester Himes

1997 • 464 Pages • 1.72 MB

Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL2(R)

2012 • 207 Pages • 1.35 MB

Transformations in Technology, Transformations in Work

2016 • 224 Pages • 8.47 MB

2016 Coexistence of multiple coronaviruses in several bat colonies in an abandoned mineshaft

2016 • 10 Pages • 664.3 KB

Electoral Systems and Political Transformation in Post-Communist Europe (One Europe or Several?)

2004 • 225 Pages • 655 KB

Several Complex Variables II: Function Theory in Classical Domains Complex Potential Theory

1994 • 266 Pages • 7.03 MB

Transformations in Metals

2007 • 415 Pages • 4.01 MB

Transformations and Projections in Computer Graphics

2007 • 283 Pages • 2.81 MB