Half-Order Differentials on Riemann Surfaces
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LOOKING BACKWARD: from EULER to RIEMANN 11 at the Age of 24
LOOKING BACKWARD: FROM EULER TO RIEMANN ATHANASE PAPADOPOULOS Il est des hommes auxquels on ne doit pas adresser d’´eloges, si l’on ne suppose pas qu’ils ont le goˆut assez peu d´elicat pour goˆuter les louanges qui viennent d’en bas. (Jules Tannery, [241] p. 102) Abstract. We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian integrals, the hypergeometric series, the zeta function, topology, differential geometry, integration, and the notion of space. We shall see that among Riemann’s predecessors in all these fields, one name occupies a prominent place, this is Leonhard Euler. The final version of this paper will appear in the book From Riemann to differential geometry and relativity (L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017. AMS Mathematics Subject Classification: 01-02, 01A55, 01A67, 26A42, 30- 03, 33C05, 00A30. Keywords: Bernhard Riemann, function of a complex variable, space, Rie- mannian geometry, trigonometric series, zeta function, differential geometry, elliptic integral, elliptic function, Abelian integral, Abelian function, hy- pergeometric function, topology, Riemann surface, Leonhard Euler, space, integration. Contents 1. Introduction 2 2. Functions 10 3. Elliptic integrals 21 arXiv:1710.03982v1 [math.HO] 11 Oct 2017 4. Abelian functions 30 5. Hypergeometric series 32 6. The zeta function 34 7. On space 41 8. Topology 47 9. Differential geometry 63 10. Trigonometric series 69 11. Integration 79 12. Conclusion 81 References 85 Date: October 12, 2017. -
Dynamical Systems/Syst`Emes Dynamiques SIMPLE
Dynamical systems/Systemes` dynamiques SIMPLE EXPONENTIAL ESTIMATE FOR THE NUMBER OF REAL ZEROS OF COMPLETE ABELIAN INTEGRALS Dmitri Novikov, Sergeı˘ Yakovenko The Weizmann Institute of Science (Rehovot, Israel) Abstract. We show that for a generic real polynomial H = H(x, y) and an arbitrary real differential 1-form ω = P (x, y) dx + Q(x, y) dy with polynomial coefficients of degree 6 d, the number of ovals of the foliation H = const, which yield the zero value of the complete H Abelian integral I(t) = H=t ω, grows at most as exp OH (d) as d → ∞, where OH (d) depends only on H. This result essentially improves the previous double exponential estimate obtained in [1] for the number of zero ovals on a bounded distance from singular level curves of H. La borne simplement exponentielle pour le nombre de zeros´ reels´ isoles´ des integrales completes` abeliennes´ R´esum´e. Soit H = H(x, y) un polynˆome r´eel de deux variables et ω = P (x, y) dx + Q(x, y) dy une forme diff´erentielle quelconque avec des coefficients polynomiaux r´eels de degr´e d. Nous montrons que le nombre des ovales (c’est-a-dire les composantes compactes connexes) des courbes de niveau H = const, telles que l’int´egrale de la forme s’annule, est au plus exp OH (d) quand d → ∞, o`u OH (d) ne d´epend que du polynˆome H. Ce th´eor`emeam´eliore de mani`ere essentielle la borne doublement exponentielle obtenue dans [1]. Nous ´etablissons aussi les bornes sup´erieures exponentielles similaires pour le nom- bre de z´eros isol´espour les enveloppes polynomiales des ´equations diff´erentielles ordinaires lin´eaires irr´eductibles ou presque irr´eductibles. -
Projective and Conformal Schwarzian Derivatives and Cohomology of Lie
Projective and Conformal Schwarzian Derivatives and Cohomology of Lie Algebras Vector Fields Related to Differential Operators Sofiane Bouarroudj Department of Mathematics, U.A.E. University, Faculty of Science P.O. Box 15551, Al-Ain, United Arab Emirates. e-mail:bouarroudj.sofi[email protected] arXiv:math/0101056v2 [math.DG] 3 Apr 2004 1 Abstract Let M be either a projective manifold (M, Π) or a pseudo-Riemannian manifold (M, g). We extend, intrinsically, the projective/conformal Schwarzian derivatives that we have introduced recently, to the space of differential op- erators acting on symmetric contravariant tensor fields of any degree on M. As operators, we show that the projective/conformal Schwarzian derivatives depend only on the projective connection Π and the conformal class [g] of the metric, respectively. Furthermore, we compute the first cohomology group of Vect(M) with coefficients into the space of symmetric contravariant tensor fields valued into δ-densities as well as the corresponding relative cohomology group with respect to sl(n + 1, R). 1 Introduction The investigation of invariant differential operators is a famous subject that have been intensively investigated by many authors. The well-known invariant operators and more studied in the literature are the Schwarzian derivative, the power of the Laplacian (see [13]) and the Beltrami operator (see [2]). We have been interested in studying the Schwarzian derivative and its relation to the geometry of the space of differential operators viewed as a module over the group of diffeomorphisms in the series of papers [4, 8, 9]. As a reminder, the classical expression of the Schwarzian derivative of a diffeomorphism f is: f ′′′ 3 f ′′ 2 − · (1.1) f ′ 2 f ′ The two following properties of the operator (1.1) are the most of interest for us: (i) It vanishes on the M¨obius group PSL(2, R) – here the group SL(2, R) acts locally on R by projective transformations. -
The Riemann Mapping Theorem Christopher J. Bishop
The Riemann Mapping Theorem Christopher J. Bishop C.J. Bishop, Mathematics Department, SUNY at Stony Brook, Stony Brook, NY 11794-3651 E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary: 30C35, Secondary: 30C85, 30C62 Key words and phrases. numerical conformal mappings, Schwarz-Christoffel formula, hyperbolic 3-manifolds, Sullivan’s theorem, convex hulls, quasiconformal mappings, quasisymmetric mappings, medial axis, CRDT algorithm The author is partially supported by NSF Grant DMS 04-05578. Abstract. These are informal notes based on lectures I am giving in MAT 626 (Topics in Complex Analysis: the Riemann mapping theorem) during Fall 2008 at Stony Brook. We will start with brief introduction to conformal mapping focusing on the Schwarz-Christoffel formula and how to compute the unknown parameters. In later chapters we will fill in some of the details of results and proofs in geometric function theory and survey various numerical methods for computing conformal maps, including a method of my own using ideas from hyperbolic and computational geometry. Contents Chapter 1. Introduction to conformal mapping 1 1. Conformal and holomorphic maps 1 2. M¨obius transformations 16 3. The Schwarz-Christoffel Formula 20 4. Crowding 27 5. Power series of Schwarz-Christoffel maps 29 6. Harmonic measure and Brownian motion 39 7. The quasiconformal distance between polygons 48 8. Schwarz-Christoffel iterations and Davis’s method 56 Chapter 2. The Riemann mapping theorem 67 1. The hyperbolic metric 67 2. Schwarz’s lemma 69 3. The Poisson integral formula 71 4. A proof of Riemann’s theorem 73 5. Koebe’s method 74 6. -
Schwarzian Derivative Related to Modules
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Sharp Bounds on the Higher Order Schwarzian Derivatives for Janowski Classes
Article Sharp Bounds on the Higher Order Schwarzian Derivatives for Janowski Classes Nak Eun Cho 1, Virendra Kumar 2,* and V. Ravichandran 3 1 Department of Applied Mathematics, Pukyong National University, Busan 48513, Korea; [email protected] 2 Department of Mathematics, Ramanujan college, University of Delhi, H-Block, Kalkaji, New Delhi 110019, India 3 Department of Mathematics, National Institute of Technology, Tiruchirappalli, Tamil Nadu 620015, India; [email protected]; [email protected] * Correspondence: [email protected] Received: 25 July 2018; Accepted: 14 August 2018; Published: 18 August 2018 Abstract: Higher order Schwarzian derivatives for normalized univalent functions were first considered by Schippers, and those of convex functions were considered by Dorff and Szynal. In the present investigation, higher order Schwarzian derivatives for the Janowski star-like and convex functions are considered, and sharp bounds for the first three consecutive derivatives are investigated. The results obtained in this paper generalize several existing results in this direction. Keywords: higher order Schwarzian derivatives; Janowski star-like function; Janowski convex function; bound on derivatives MSC: 30C45; 30C50; 30C80 1. Introduction Let A be the class of analytic functions defined on the unit disk D := fz 2 C : jzj < 1g and having the form: 2 3 f (z) = z + a2z + a3z + ··· . (1) The subclass of A consisting of univalent functions is denoted by S. An analytic function f is subordinate to another analytic function g if there is an analytic function w with jw(z)j ≤ jzj and w(0) = 0 such that f (z) = g(w(z)), and we write f ≺ g. If g is univalent, then f ≺ g if and only if ∗ f (0) = g(0) and f (D) ⊆ g(D). -
Abelian Integrals and the Genesis of Abel's Greatest Discovery
Abelian integrals and the genesis of Abel's greatest discovery Christian F. Skau Department of Mathematical Sciences, Norwegian University of Science and Technology March 20, 2020 Draft P(x) R(x) = Q(x) 3x 7 − 18x 6 + 46x 5 − 84x 4 + 127x 3 − 122x 2 + 84x − 56 = (x − 2)3(x + i)(x − i) 4x 4 − 12x 3 + 7x 2 − x + 1 = 3x 2 + 7 + (x − 2)3(x + i)(x − i) (−1) 3 4 i=2 (−i=2) = 3x 2 + 7 + + + + + (x − 2)3 (x − 2)2 x − 2 x + i x − i Draft Z 1=2 −3 R(x) dx = x 3 + 7x + + + 4 log (x − 2) (x − 2)2 x − 2 i −i + log (x + i) + log (x − i) 2 2 i −i = Re(x) + 4 log (x − 2) + log (x + i) + log (x − i) 2 2 The integral of a rational function R(x) is a rational function Re(x) plus a sum of logarithms of the form log(x + a), a 2 C, multiplied with constants. Draft eix − e−ix 1 p sin x = arcsin x = log ix + 1 − x 2 2i i Trigonometric functions and the inverse arcus-functions can be expressed by (complex) exponential and logarithmic functions. Draft A rational function R(x; y) of x and y is of the form P(x; y) R(x; y) = ; Q(x; y) where P and Q are polynomials in x and y. So, P i j aij x y R(x; y) = P k l bkl x y Similarly we define a rational function of x1; x2;:::; xn. -
The Historical Development of Algebraic Geometry∗
The Historical Development of Algebraic Geometry∗ Jean Dieudonn´e March 3, 1972y 1 The lecture This is an enriched transcription of footage posted by the University of Wis- consin{Milwaukee Department of Mathematical Sciences [1]. The images are composites of screenshots from the footage manipulated with Python and the Python OpenCV library [2]. These materials were prepared by Ryan C. Schwiebert with the goal of capturing and restoring some of the material. The lecture presents the content of Dieudonn´e'sarticle of the same title [3] as well as the content of earlier lecture notes [4], but the live lecture is natu- rally embellished with different details. The text presented here is, for the most part, written as it was spoken. This is why there are some ungrammatical con- structions of spoken word, and some linguistic quirks of a French speaker. The transcriber has taken some liberties by abbreviating places where Dieudonn´e self-corrected. That is, short phrases which turned out to be false starts have been elided, and now only the subsequent word-choices that replaced them ap- pear. Unfortunately, time has taken its toll on the footage, and it appears that a brief portion at the beginning, as well as the last enumerated period (VII. Sheaves and schemes), have been lost. (Fortunately, we can still read about them in [3]!) The audio and video quality has contributed to several transcription errors. Finally, a lack of thorough knowledge of the history of algebraic geometry has also contributed some errors. Contact with suggestions for improvement of arXiv:1803.00163v1 [math.HO] 1 Mar 2018 the materials is welcome. -
The Chazy XII Equation and Schwarz Triangle Functions
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 095, 24 pages The Chazy XII Equation and Schwarz Triangle Functions Oksana BIHUN and Sarbarish CHAKRAVARTY Department of Mathematics, University of Colorado, Colorado Springs, CO 80918, USA E-mail: [email protected], [email protected] Received June 21, 2017, in final form December 12, 2017; Published online December 25, 2017 https://doi.org/10.3842/SIGMA.2017.095 Abstract. Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120{348] showed that the Chazy XII equation y000 − 2yy00 + 3y02 = K(6y0 − y2)2, K 2 C, is equivalent to a projective-invariant equation for an affine connection on a one-dimensional complex manifold with projective structure. By exploiting this geometric connection it is shown that the Chazy XII solution, for certain values of K, can be expressed as y = a1w1 +a2w2 +a3w3 where wi solve the generalized Darboux{Halphen system. This relationship holds only for certain values of the coefficients (a1; a2; a3) and the Darboux{Halphen parameters (α; β; γ), which are enumerated in Table2. Consequently, the Chazy XII solution y(z) is parametrized by a particular class of Schwarz triangle functions S(α; β; γ; z) which are used to represent the solutions wi of the Darboux{Halphen system. The paper only considers the case where α + β +γ < 1. The associated triangle functions are related among themselves via rational maps that are derived from the classical algebraic transformations of hypergeometric functions. The Chazy XII equation is also shown to be equivalent to a Ramanujan-type differential system for a triple (P;^ Q;^ R^). -
Schwarzian Derivatives of Topologically Finite Meromorphic Functions
Annales Academi½ Scientiarum Fennic½ Mathematica Volumen 27, 2002, 215{220 SCHWARZIAN DERIVATIVES OF TOPOLOGICALLY FINITE MEROMORPHIC FUNCTIONS Adam Lawrence Epstein University of Warwick, Mathematics Institute Coventry CV4 7AL, United Kingdom; [email protected] Abstract. Using results from dynamics, we prove sharp upper bounds on the local vanishing order of the nonlinearity and Schwarzian derivative for topologically ¯nite meromorphic functions. The entire functions f: C C with rational nonlinearity ! f 00 Nf = f 0 and more generally the meromorphic functions f: C C with rational Schwar- zian derivative ! 1 2 S = N0 N b f f ¡ 2 f were discussed classically by the brothers Nevanlinna and a number of other re- searchers. These functions admit a purely topological characterization (for details, see [6, pp. 152{153], [7, pp. 391{393] and [11, pp. 298{303]). We show here that a topological ¯niteness property enjoyed by a considerably larger class of functions guarantees bounds on the local vanishing orders of Nf and Sf . For meromorphic f: C C, let S(f) denote the set of all singular (that is, critical or asymptotic) ! values in C. It follows by elementary covering space theory that #S(f) 2, pro- vided thatb f is not a MÄobius transformation; note further that S(f¸) for any 1 2 entire f: Cb C, provided that f is not a±ne. Our main result is the following: ! Theorem 1. A. If f: C C is meromorphic but not a MÄobius transfor- mation then ord S #S(f) !2; consequently ord N #S(f) 1, for every ³ f · ¡ ³ f · ¡ ³ C. -
Maps with Polynomial Schwarzian Derivative Annales Scientifiques De L’É.N.S
ANNALES SCIENTIFIQUES DE L’É.N.S. ROBERT L. DEVANEY LINDA KEEN Dynamics of meromorphic maps : maps with polynomial schwarzian derivative Annales scientifiques de l’É.N.S. 4e série, tome 22, no 1 (1989), p. 55-79 <http://www.numdam.org/item?id=ASENS_1989_4_22_1_55_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1989, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. sclent. EC. Norm. Sup., 46 serie, t. 22, 1989, p. 55 a 79. DYNAMICS OF MEROMORPHIC MAPS: MAPS WITH POLYNOMIAL SCHWARZIAN DERIVATIVE BY ROBERT L. DEVANEY AND LINDA KEEN ABSTRACT. - We investigate the dynamics of a class of meromorphic functions, namely those with polynomial Schwarzian derivative. We show that certain of these maps have dynamics similar to rational maps while others resemble entire functions. We use classical results of Nevanlinna and Hille to describe certain aspects of the topological structure and dynamics of these maps on their Julia sets. Several examples from the families z -> X tan z, z ->• ez/(^z+e-z), and z -^'ke^^—e~z) are discussed in detail. There has been a resurgence of interest in the past decade in the dynamics of complex analytic functions. -
Arxiv:2101.09561V2 [Math.CV] 4 Feb 2021
QUASICONFORMAL EXTENSION FOR HARMONIC MAPPINGS ON FINITELY CONNECTED DOMAINS IASON EFRAIMIDIS Abstract. We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal extension to the whole plane if its Schwarzian derivative is small. We also make the observation that a univalence criterion for harmonic mappings holds on uniform domains. 1. Introduction Let f be a harmonic mapping in a planar domain D and let ! = fz=fz be its dilatation. According to Lewy's theorem the mapping f is locally univalent 2 2 if and only if its Jacobian Jf = jfzj − jfzj does not vanish. Duren's book [4] contains valuable information about the theory of planar harmonic mappings. The Schwarzian derivative of f was defined by Hern´andezand Mart´ın[8] as 1 2 Sf = ρzz − 2 (ρz) ; where ρ = log Jf : (1) When f is holomorphic this reduces to the classical Schwarzian derivative. An- other definition, introduced by Chuaqui, Duren and Osgood [3], applies to har- monic mappings which admit a lift to a minimal surface via the Weierstrass- Enneper formulas. However, focusing on the planar theory in this note we adopt the definition (1). We assume that CnD contains at least three points, so that D is equipped with the hyperbolic metric, defined by jdzj λ π(z)jπ0(z)jjdzj = λ (z)jdzj = ; z 2 ; D D 1 − jzj2 D where D is the unit disk and π : D ! D is a universal covering map. The size of the Schwarzian derivative of a mapping f in D is measured by the norm −2 kSf kD = sup λD(z) jSf (z)j: arXiv:2101.09561v2 [math.CV] 4 Feb 2021 z2D A domain D in C is a K-quasidisk if it is the image of the unit disk under a K-quasiconformal self-map of C, for some K ≥ 1.