MODULI OF FAMILIES OF CURVES FOR CONFORMAL AND QUASICONFORMAL MAPPINGS
CONTENTS 1 INTRODUCTION 1 2 MODULI OF FAMILIES OF CURVES AND EXTREMAL PARTITIONS 7 2 1 SIMPLE DE NITION AND PROPERTIES OF THE MODULUS 7 2 1 1 DE NITION 7 2 1 2 PROPERTIES 8 2 1 3 EXAMPLES 10 2 1 4 GR OTZSCH LEMMAS 13 2 1 5 EXERCISES 15 2 2 REDUCED MODULI AND CAPACITY 15 2 2 1 REDUCED MODULUS 15 2 2 2 CAPACITY AND TRANS NITE DIAMETER 17 2 2 3 DIGONS TRIANGLES AND THEIR REDUCED MODULI 19 2 3 ELLIPTIC FUNCTIONS AND INTEGRALS 23 2 3 1 ELLIPTIC FUNCTIONS 23 2 3 2 ELLIPTIC INTEGRALS AND JACOBI S FUNCTIONS 25 2 4 SOME FREQUENTLY USED MODULI 30 2 4 1 MODULI OF DOUBLY CONNECTED DOMAINS 30 2 4 2 MODULI OF QUADRILATERALS 32 2 4 3 REDUCED MODULI 32 2 4 4 REDUCED MODULI OF DIGONS 33 2 5 SYMMETRIZATION AND POLARIZATION 34 2 5 1 CIRCULAR SYMMETRIZATION 34 2 5 2 POLARIZATION 35 2 6 QUADRATIC DI ERENTIALS ON RIEMANN SURFACES 37 2 6 1 RIEMANN SURFACES 37 2 6 2 QUADRATIC DI ERENTIALS 39 2 6 3 LOCAL TRAJECTORY STRUCTURE 40 2 6 4 TRAJECTORY STRUCTURE IN THE LARGE 43 2 7 FREE FAMILIES OF HOMOTOPY CLASSES OF CURVES AND EXTREMAL PAR TITIONS 46 2 7 1 THE CASE OF RING DOMAINS AND QUADRANGLES 46 2 7 2 THE CASE OF CIRCULAR STRIP DOMAINS AND TRIANGLES 50 2 7 3 CONTINUOUS AND DI ERENTIABLE MODULI 54 3 MODULI IN EXTREMAL PROBLEMS FOR CONFORMAL MAPPING 57 3 1 CLASSICAL EXTREMAL PROBLEMS FOR UNIVALENT FUNCTIONS 57 3 1 1 KOEBE SET GROWTH DISTORTION 57 3 1 2 LOWER BOUNDARY CURVE FOR THE RANGE OF F Z F Z 60 3 1 3 SPECIAL MODULI 61 3 1 4 UPPER BOUNDARY CURVE FOR THE RANGE OF F Z F Z 66 3 2 TWO POINT DISTORTION FOR UNIVALENT FUNCTIONS 70 3 2 1 LOWER BOUNDARY CURVE FOR THE RANGE OF F R1 F R2 IN SR 70 3 2 2 SPECIAL MODULI 71 3 2 3 UPPER BOUNDARY CURVE FOR THE RANGE OF F R1 F R2 IN SR 80 3 2 4 UPPER BOUNDARY CURVE FOR THE RANGE OF F R1 F R2 IN S 85 3 3 BOUNDED UNIVALENT FUNCTIONS 86 3 3 1 ELEMENTARY ESTIMATES 87 3 3 2 BOUNDARY CURVE FOR THE RANGE OF F Z F Z IN BS B 89 3 4 MONTEL FUNCTIONS 94 3 4 1 COVERING THEOREMS 95 3 4 2 DISTORTION AT THE POINTS OF NORMALIZATION 98 3 4 3 THE RANGE OF F R F R IN MR 108 3 5 UNIVALENT FUNCTIONS WITH THE ANGULAR DERIVATIVES 120 3 5 1 ESTIMATES OF THE ANGULAR DERIVATIVES 120 3 5 2 THE RANGE OF F R F 0 125 4 MODULI IN EXTREMAL PROBLEMS FOR QUASICONFORMAL MAPPING141 4 1 GENERAL INFORMATION AND SIMPLE EXTREMAL PROBLEMS 141 4 1 1 QUASICONFORMAL MAPPINGS OF RIEMANN SURFACES 141 4 1 2 GROWTH AND H OLDER CONTINUITY 145 4 1 3 QUASICONFORMAL MOTION OF A QUADRUPLE OF POINTS 148 4 2 TWO POINT DISTORTION FOR QUASICONFORMAL MAPS OF THE PLANE 151 4 2 1 SPECIAL DI ERENTIALS AND EXTREMAL PARTITIONS 153 4 2 2 QUASISYMMETRIC FUNCTIONS AND THE EXTREMAL MAPS 154 4 2 3 BOUNDARY PARAMETERIZATION 157 4 2 4 THE CLASS QK ESTIMATIONS OF FUNCTIONALS 159 4 2 5 CONCLUSIONS AND UNSOLVED PROBLEMS 166 4 3 TWO POINT DISTORTION FOR QUASICONFORMAL MAPS OF THE UNIT DISK 168 4 3 1 SPECIAL DI ERENTIALS AND EXTREMAL PARTITIONS 168 4 3 2 EXTREMAL PROBLEMS 172 5 MODULI ON TEICHM ULLER SPACES 175 5 1 SOME INFORMATION ON TEICHM ULLER SPACES 175 5 2 MODULI ON TEICHM ULLER SPACES 177 5 2 1 VARIATIONAL FORMULAE 178 5 2 2 THREE LEMMAS 184 5 3 HARMONIC PROPERTIES OF THE MODULI 187 5 4 DESCRIPTIONS OF THE TEICHM ULLER METRIC 190 5 5 INVARIANT METRICS 193 1 INTRODUCTION AN IMPORTANT PART OF COMPLEX ANALYSIS IS EXTREMAL PROBLEMS RST OF ALL ESTIMATION OF FUNCTIONALS AND DESCRIPTION OF THE RANGES OF SYSTEM OF FUNCTION ALS IN VARIOUS COMPACT CLASSES OF CONFORMAL OR QUASICONFORMAL MAPS MANY POWERFUL METHODS HAVE BEEN DEVELOPED TO SOLVE THE PROBLEMS FOR CONFORMAL MAPS WE MENTION HERE THE AREA PRINCIPLE 48 88 THE VARIATIONAL METHOD OF GOLUZIN SCHI ER 48 118 120 THE L OWNER KUFAREV PARAMETRIC METHOD 4 56 94 112 THE LATTER HAS ALLOWED NALLY TO SOLVE THE BIEBERBACH CONJECTURE SEE THE HISTORY E G IN 21 111 ONE OF THE CRUCIAL PROBLEMS IS THE UNIQUENESS OF THE EXTREMAL FUNCTIONS FOR A GIVEN FUNCTIONAL THE COMPLETE SOLUTION OF SUCH A PROBLEM IS GIVEN BY THE MODULUS METHOD THAT USES EITHER VARIATIONAL PRINCIPLES OR TOPOLOGICAL GEOMETRIC ONES THE METHOD OF EXTREMAL METRICS EXTREMAL LENGTH METHOD GOES BACK TO THE STRIP METHOD BY H GR OTZSCH 53 54 H GR OTZSCH RST USED IT AS A METHOD IN THE THEORY OF UNIVALENT FUNCTIONS HE STATES THAT ITS USE WAS SUGGESTED TO HIM BY THE WORK OF FABER SEE 64 PAGE 7 LATER L V AHLFORS AND A BEURLING 9 12 INTRODUCED THE NOTION OF THE EXTREMAL LENGTH THE RECIPROCAL OF THE MODULUS THAT GAVE RISE TO THE ACTIVE DEVELOPMENT OF THE METHOD THE GREAT CONTRIBUTION TO THE SUBJECT HAS BEEN MADE BY J JENKINS 63 64 AND K STREBEL 141 WHO CONNECTED THE MODULUS PROBLEM WITH THE PROBLEM OF THE EXTREMAL PARTITIONING OF A RIEMANN SURFACE AND PROVED THE EXISTENCE OF THE EXTREMAL METRIC BY SCHI ER S VARIATIONS 119 FURTHER DEVELOPMENT AND GENERALIZATION OF THE MODULUS WERE MADE BY G V KUZ MINA 78 NOW ON THE MODULUS METHOD THE EXTREMAL LENGTH METHOD IN AHLFORS TERMINOLOGY IS A POWERFUL TOOL FOR THE SOLUTION OF PROBLEMS OF VARIOUS TYPES IT TURNS OUT TO BE RATHER UNIVERSAL AND PROVIDES EASILY AND NATURALLY THE UNIQUENESS OF SOLUTION IN MANY EXTREMAL PROBLEMS OF CONFORMAL MAPPING WHAT IS COMPLICATE ON APPLYING OTHER METHODS IT IS IMPORTANT THAT THE MODULUS BECAME A SOURCE OF THE DE NITION OF QUASICONFORMAL MAPS THE THEORY OF QUASICONFORMAL MAPPING EMERGED AT THE BEGINNING OF THE TWENTIETH CENTURY AT THAT TIME THESE MAPS AROSE BY GEOMETRIC REASONS BASED ON THE WORKS OF H GR OTZSCH 53 54 WHO INTRODUCED SO CALLED REG ULAR QUASICONFORMAL MAPS AND THE NOTION OF THE EXTREMAL LENGTH SUGGESTED BY L V AHLFORS AND A BEURLING AND ON THE OTHER HAND AS SOLUTIONS OF A SPECIAL TYPE OF ELLIPTIC SYSTEMS OF DI ERENTIAL EQUATIONS IN THE WORKS BY 1