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Quasiconformal Mappings, Complex Dynamics and Moduli Spaces Glutsyuk A

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Quasiconformal Mappings, Complex Dynamics and Moduli Spaces Glutsyuk A Syllabus 1. Course Description a. Title of a Course: Quasiconformal mappings, complex dynamics and moduli spaces Glutsyuk A. b. Pre-requisites: complex analysis in one variable c. Course Type: elective, optional d. Abstract: Poincaré--Koebe Uniformization Theorem is a fundamental theorem of geometry saying that each simply-connected Riemann surface is conformally-equivalent to either the Riemann sphere, or the complex line, or unit disk. The quasiconformal mapping theory was founded by H.Grötzsch, M.A.Lavrentiev and C.Morrey Jr. in 1920-1930-ths and later developed by L.Ahlfors and L.Bers. It extends the Uniformization Theorem to non-constant and even discontinuous complex structures. It has many important applications in various branches of mathematics: complex dynamics, Kleinian groups, Teichmüller theory, moduli spaces, geometry...Including the fundamental breakthrough in rational dynamics on the Riemann sphere: the famous Sullivan's No Wanderind Domain Theorem on absence of wandering components of the Fatou set (1980). We will study basic quasiconformal mapping theory and give a proof of its main theorem, the Measurable Riemann Mapping Theorem. Then we pass to applications to dynamics of rational mappings of the Riemann sphere and Kleinian groups: finitely-generated discrete groups of conformal automorphisms of the Riemann sphere. We will prove the No Wandering Domain Theorem and discuss structural stability in rational dynamics, and the analogous Ahlfors' Finiteness Theorem in Kleinian groups. Afterwards we will pass to Teichmüller theory, which deals with the space of different complex structures on a topologically marked closed surface, and study its quotient space: the moduli space of Riemann surfaces. 2. Learning Objectives: Learning basic quasiconformal mapping theory, proof of Measurable Riemann Mapping Theorem, studying applications in complex dynamics, Kleinian groups amd moduli spaces, proofs of Sullivan's No Wandering Domain Theorem and Ahlfors' Finiteness Theorem, studying the geometry of Teichmüller and moduli spaces of Riemann surfaces using quasiconformal mappings, studying Fuchsian groups and the parametrization of the moduli space of Riemann surfaces coming from Fuchsian groups. 3. Learning Outcomes: As a result of learning the subject, a student is supposed to: - know main definitions and theorems in the theory of quasiconformal mappings, including Measurable Riemann Mapping Theorem, Grötzsch inequality; - know main application of the quasiconformal mapping theory in complex dynamics and Kleinian groups, including first of all Sullivan's No Wandering Domain Theorem and Ahlfors' Finiteness Theorem; - know how to define complex structure and coordinates on Teichmüller and moduli spaces, know Bers' Simultaneous Uniformization Theorem. 4. Course Plan Chapter 1. Basic quasiconformal mapping theory. 1.1. Poincaré—Koebe Uniformization Theorem. Quasiconformal mappings. Measurable Riemann Mapping Theorem. 1.2. Proof of Measurable Riemann Mapping theorem for smooth complex structures on two- torus. 1.3. Grötzsch inequality. Proof of the theorem in the general case. Chapter 2. Application in complex dynamics. 2.1. Rational iterations on Riemann sphere. Fatou set. Examples. Periodic components of the Fatou set. Sullivan's No Wandering Domain Theorem. 2.2. Proof of Sullivan's theorem. 2.3. Structural stability of rational mappings and holomorphic motions. Mane-Sad-Sullivan Theorem on structural stability. Holomorphic motions: Slodkowski's theorem (without proof). Chapter 3. Applications to Kleinian groups. 3.1. Kleinian groups. Ahlfors' Finiteness Theorem, with proof. Ahlfors' Measure Conjecture (Agol-Calegari-Gabai theorem, without proof). 3.2. Fuchsian and quasifuchsian groups. Bers Simultaneous Uniformization Theorem. Chapter 4. Teichmüller and moduli spaces. 4.1.Teichmüller distance and space. Moduli space. Teichmüller geodesics. 4.2.Complex structure and coordinates on Teichmüller and moduli space. Coordinates coming from Fuchsian groups. 4.3.Tangent space to moduli space, quadratic differentials. 5. Reading List [1] L.Ahlfors. Lectures on quasiconformal mappings. Moscow, Mir, 1969 (in Russian). English version, second edition, appeared in AMS University Lecture Series, 38 (2006), American Mathematical Society, Rhode Island, Providence. [2] J.Hubbard. Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Matrix Editions, 2006. [3] C.McMullen. Riemann surfaces, dynamics and geometry. Course notes: http://www.math.harvard.edu/~ctm/home/text/class/harvard/275/09/html/base/rs/rs.pdf [4] A.Glutsyuk. Simple proof of Uniformization Theorems. Fields Institute Communications, 53 (2008), 125--143. 6. Grading System Colloquium (middle partial examination): a student should show that he can distinguish whether a given mapping is quasiconformal or not, find the image of a given almost complex structure by a given quasiconformal mapping, find Fatou and Julia set of a given simple rational function, find whether a given simple rational function is structurally stable, - use construction of invariant almost complex structures in order to construct a rational mapping with given dynamical properties. Final examination: a student should be able to show that he masters the skills listed above and in Section 3. 7. Guidelines for Knowledge Assessment For homework and colloquium: 7.1. Find out whether a given mapping is quasiconformal and find its dilatation. 7.2. Find the image of a given linear complex (almost complex) structure under a linear (respectively, quasiconformal) transformation. 7.3. Prove an analogue of Grötzsch inequality for rectangles replaced by tori. 7.4. Construct a quasiconformal mapping transforming a given almost complex structure with a one-parametric symmetry group to the standard one. 7.5. Find the Fatou set of a given rational function. 7.6. Find out whether a given rational function is structurally stable. 7.7. Construct an explicit example of Fuchsian group uniformizing a Riemann surface of genus two. 7.8. Consider the Teichmüller space of the Riemann sphere with a finite number of marked points. Find its dimension and biholomorphic type. Introduce cordinates on it. 7.9. Study the subspace of the Teichmüller space corresponding to topologically marked hyperelliptic Riemann surfaces. Introduce coordinates on it. 7.10. Study quadratic differentials on a hyperelliptic Riemann surface. Calculate the dimension of the space of those of them that correspond to tangent vectors to the Teichmüller space. 8. Methods of Instruction: the system of problem lists. 9. Special Equipment and Software Support (if required): no. .
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