Calabi Conjecture
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Millennium Prize for the Poincaré
FOR IMMEDIATE RELEASE • March 18, 2010 Press contact: James Carlson: [email protected]; 617-852-7490 See also the Clay Mathematics Institute website: • The Poincaré conjecture and Dr. Perelmanʼs work: http://www.claymath.org/poincare • The Millennium Prizes: http://www.claymath.org/millennium/ • Full text: http://www.claymath.org/poincare/millenniumprize.pdf First Clay Mathematics Institute Millennium Prize Announced Today Prize for Resolution of the Poincaré Conjecture a Awarded to Dr. Grigoriy Perelman The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture. The citation for the award reads: The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman. The Poincaré conjecture is one of the seven Millennium Prize Problems established by CMI in 2000. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude. The award of the Millennium Prize to Dr. Perelman was made in accord with their governing rules: recommendation first by a Special Advisory Committee (Simon Donaldson, David Gabai, Mikhail Gromov, Terence Tao, and Andrew Wiles), then by the CMI Scientific Advisory Board (James Carlson, Simon Donaldson, Gregory Margulis, Richard Melrose, Yum-Tong Siu, and Andrew Wiles), with final decision by the Board of Directors (Landon T. -
The Work of Grigory Perelman
The work of Grigory Perelman John Lott Grigory Perelman has been awarded the Fields Medal for his contributions to geom- etry and his revolutionary insights into the analytical and geometric structure of the Ricci flow. Perelman was born in 1966 and received his doctorate from St. Petersburg State University. He quickly became renowned for his work in Riemannian geometry and Alexandrov geometry, the latter being a form of Riemannian geometry for metric spaces. Some of Perelman’s results in Alexandrov geometry are summarized in his 1994 ICM talk [20]. We state one of his results in Riemannian geometry. In a short and striking article, Perelman proved the so-called Soul Conjecture. Soul Conjecture (conjectured by Cheeger–Gromoll [2] in 1972, proved by Perelman [19] in 1994). Let M be a complete connected noncompact Riemannian manifold with nonnegative sectional curvatures. If there is a point where all of the sectional curvatures are positive then M is diffeomorphic to Euclidean space. In the 1990s, Perelman shifted the focus of his research to the Ricci flow and its applications to the geometrization of three-dimensional manifolds. In three preprints [21], [22], [23] posted on the arXiv in 2002–2003, Perelman presented proofs of the Poincaré conjecture and the geometrization conjecture. The Poincaré conjecture dates back to 1904 [24]. The version stated by Poincaré is equivalent to the following. Poincaré conjecture. A simply-connected closed (= compact boundaryless) smooth 3-dimensional manifold is diffeomorphic to the 3-sphere. Thurston’s geometrization conjecture is a far-reaching generalization of the Poin- caré conjecture. It says that any closed orientable 3-dimensional manifold can be canonically cut along 2-spheres and 2-tori into “geometric pieces” [27]. -
Kähler-Einstein Metrics and Algebraic Geometry
Current Developments in Mathematics, 2015 K¨ahler-Einstein metrics and algebraic geometry Simon Donaldson Abstract. This paper is a survey of some recent developments in the area described by the title, and follows the lines of the author’s lecture in the 2015 Harvard Current Developments in Mathematics meeting. The main focus of the paper is on the Yau conjecture relating the ex- istence of K¨ahler-Einstein metrics on Fano manifolds to K-stability. We discuss four different proofs of this, by different authors, which have ap- peared over the past few years. These involve an interesting variety of approaches and draw on techniques from different fields. Contents 1. Introduction 1 2. K-stability 3 3. Riemannian convergence theory and projective embeddings 6 4. Four proofs 11 References 23 1. Introduction General existence questions involving the Ricci curvature of compact K¨ahler manifolds go back at least to work of Calabi in the 1950’s [11], [12]. We begin by recalling some very basic notions in K¨ahler geometry. • All the K¨ahler metrics in a given cohomology class can be described in terms of some fixed reference metric ω0 and a potential function, that is (1) ω = ω0 + i∂∂φ. • A hermitian holomorphic line bundle over a complex manifold has a unique Chern connection compatible with both structures. A Her- −1 n mitian metric on the anticanonical line bundle KX =Λ TX is the same as a volume form on the manifold. When this volume form is derived from a K¨ahler metric the curvature of the Chern connection c 2016 International Press 1 2 S. -
A Survey of the Elastic Flow of Curves and Networks
A survey of the elastic flow of curves and networks Carlo Mantegazza ∗ Alessandra Pluda y Marco Pozzetta y September 28, 2020 Abstract We collect and present in a unified way several results in recent years about the elastic flow of curves and networks, trying to draw the state of the art of the subject. In par- ticular, we give a complete proof of global existence and smooth convergence to critical points of the solution of the elastic flow of closed curves in R2. In the last section of the paper we also discuss a list of open problems. Mathematics Subject Classification (2020): 53E40 (primary); 35G31, 35A01, 35B40. 1 Introduction The study of geometric flows is a very flourishing mathematical field and geometric evolution equations have been applied to a variety of topological, analytical and physical problems, giving in some cases very fruitful results. In particular, a great attention has been devoted to the analysis of harmonic map flow, mean curvature flow and Ricci flow. With serious efforts from the members of the mathematical community the understanding of these topics grad- ually improved and it culminated with Perelman’s proof of the Poincare´ conjecture making use of the Ricci flow, completing Hamilton’s program. The enthusiasm for such a marvelous result encouraged more and more researchers to investigate properties and applications of general geometric flows and the field branched out in various different directions, including higher order flows, among which we mention the Willmore flow. In the last two decades a certain number of authors focused on the one dimensional analog of the Willmore flow (see [26]): the elastic flow of curves and networks. -
Non-Singular Solutions of the Ricci Flow on Three-Manifolds
COMMUNICATIONS IN ANALYSIS AND GEOMETRY Volume 7, Number 4, 695-729, 1999 Non-singular solutions of the Ricci flow on three-manifolds RICHARD S. HAMILTON Contents. 1. Classification 695 2. The lower bound on scalar curvature 697 3. Limits 698 4. Long Time Pinching 699 5. Positive Curvature Limits 701 6. Zero Curvature Limits 702 7. Negative Curvature Limits 705 8. Rigidity of Hyperbolic metrics 707 9. Harmonic Parametrizations 709 10. Hyperbolic Pieces 713 11. Variation of Area 716 12. Bounding Length by Area 720 1. Classification. In this paper we shall analyze the behavior of all non-singular solutions to the Ricci flow on a compact three-manifold. We consider only essential singularities, those which cannot be removed by rescaling alone. Recall that the normalized Ricci flow ([HI]) is given by a metric g(x,y) evolving 695 696 Richard Hamilton by its Ricci curvature Rc(x,y) with a "cosmological constant" r = r{t) representing the mean scalar curvature: !*<XfY) = 2 ^rg{X,Y) - Rc(X,Y) where -h/h- This differs from the unnormalized flow (without r) only by rescaling in space and time so that the total volume V = /1 remains constant. Definition 1.1. A non-singular solution of the Ricci flow is one where the solution of the normalized flow exists for all time 0 < t < oo, and the curvature remains bounded |jRra| < M < oo for all time with some constant M independent of t. For example, any solution to the Ricci flow on a compact three-manifold with positive Ricci curvature is non-singular, as are the equivariant solutions on torus bundles over the circle found by Isenberg and the author [H-I] which has a homogeneous solution, or the Koiso soliton on a certain four-manifold [K]; by contrast the solutions on a four-manifold with positive isotropic curvature in [H5] definitely become singular, and these singularities must be removed by surgery. -
Ricci Flow and Nonlinear Reaction--Diffusion Systems In
Ricci Flow and Nonlinear Reaction–Diffusion Systems in Biology, Chemistry, and Physics Vladimir G. Ivancevic∗ Tijana T. Ivancevic† Abstract This paper proposes the Ricci–flow equation from Riemannian geometry as a general ge- ometric framework for various nonlinear reaction–diffusion systems (and related dissipative solitons) in mathematical biology. More precisely, we propose a conjecture that any kind of reaction–diffusion processes in biology, chemistry and physics can be modelled by the com- bined geometric–diffusion system. In order to demonstrate the validity of this hypothesis, we review a number of popular nonlinear reaction–diffusion systems and try to show that they can all be subsumed by the presented geometric framework of the Ricci flow. Keywords: geometrical Ricci flow, nonlinear bio–reaction–diffusion, dissipative solitons and breathers 1 Introduction Parabolic reaction–diffusion systems are abundant in mathematical biology. They are mathemat- ical models that describe how the concentration of one or more substances distributed in space changes under the influence of two processes: local chemical reactions in which the substances are converted into each other, and diffusion which causes the substances to spread out in space. More formally, they are expressed as semi–linear parabolic partial differential equations (PDEs, see e.g., [55]). The evolution of the state vector u(x,t) describing the concentration of the different reagents is determined by anisotropic diffusion as well as local reactions: ∂tu = D∆u + R(u), (∂t = ∂/∂t), (1) arXiv:0806.2194v6 [nlin.PS] 19 May 2011 where each component of the state vector u(x,t) represents the concentration of one substance, ∆ is the standard Laplacian operator, D is a symmetric positive–definite matrix of diffusion coefficients (which are proportional to the velocity of the diffusing particles) and R(u) accounts for all local reactions. -
The Role of Partial Differential Equations in Differential Geometry
Proceedings of the International Congress of Mathematicians Helsinki, 1978 The Role of Partial Differential Equations in Differential Geometry Shing-Tung Yau In the study of geometric objects that arise naturally, the main tools are either groups or equations. In the first case, powerful algebraic methods are available and enable one to solve many deep problems. While algebraic methods are still important in the second case, analytic methods play a dominant role, especially when the defining equations are transcendental. Indeed, even in the situation where the geometric abject is homogeneous or algebraic, analytic methods often lead to important contributions. In this talk, we shall discuss a class of problems in dif ferential geometry and the analytic methods that are involved in solving such problems. One of the main purposes of differential geometry is to understand how a surface (or a generalization of it) is curved, either intrinsically or extrinsically. Naturally, the problems that are involved in studying such an object cannot be linear. Since curvature is defined by differentiating certain quantities, the equations that arise are nonlinear differential equations. In studying curved space, one of the most important tools is the space of tangent vectors to the curved space. In the language of partial differential equations, the main tool to study nonlinear equations is the use of the linearized operators. Hence, even when we are facing nonlinear objects, the theory of linear operators is unavoidable. Needless to say, we are then left with the difficult problem of how precisely a linear operator approximates a non linear operator. To illustrate the situation, we mention five important differential operators in differential geometry. -
Visualization of 2-Dimensional Ricci Flow
Pure and Applied Mathematics Quarterly Volume 9, Number 3 417—435, 2013 Visualization of 2-Dimensional Ricci Flow Junfei Dai, Wei Luo∗, Min Zhang, Xianfeng Gu and Shing-Tung Yau Abstract: The Ricci flow is a powerful curvature flow method, which has been successfully applied in proving the Poincar´e conjecture. This work introduces a series of algorithms for visualizing Ricci flows on general surfaces. The Ricci flow refers to deforming a Riemannian metric on a surface pro- portional to its Gaussian curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature on the surface converges to a con- stant function. Furthermore, during the deformation, the conformal structure of the surfaces is preserved. By using Ricci flow, any closed compact surface can be deformed into either a round sphere, a flat torus or a hyperbolic surface. This work aims at visualizing the abstract Ricci curvature flow partially us- ing the recent work of Izmestiev [8]. The major challenges are to represent the intrinsic Riemannian metric of a surface by extrinsic embedding in the three dimensional Euclidean space and demonstrate the deformation process which preserves the conformal structure. A series of rigorous and practical algorithms are introduced to tackle the problem. First, the Ricci flow is discretized to be carried out on meshes. The preservation of the conformal structure is demon- strated using texture mapping. The curvature redistribution and flow pattern is depicted as vector fields and flow fields on the surface. The embedding of convex surfaces with prescribed Riemannian metrics is illustrated by embedding meshes with given edge lengths. -
Hamilton's Ricci Flow
The University of Melbourne, Department of Mathematics and Statistics Hamilton's Ricci Flow Nick Sheridan Supervisor: Associate Professor Craig Hodgson Second Reader: Professor Hyam Rubinstein Honours Thesis, November 2006. Abstract The aim of this project is to introduce the basics of Hamilton's Ricci Flow. The Ricci flow is a pde for evolving the metric tensor in a Riemannian manifold to make it \rounder", in the hope that one may draw topological conclusions from the existence of such \round" metrics. Indeed, the Ricci flow has recently been used to prove two very deep theorems in topology, namely the Geometrization and Poincar´eConjectures. We begin with a brief survey of the differential geometry that is needed in the Ricci flow, then proceed to introduce its basic properties and the basic techniques used to understand it, for example, proving existence and uniqueness and bounds on derivatives of curvature under the Ricci flow using the maximum principle. We use these results to prove the \original" Ricci flow theorem { the 1982 theorem of Richard Hamilton that closed 3-manifolds which admit metrics of strictly positive Ricci curvature are diffeomorphic to quotients of the round 3-sphere by finite groups of isometries acting freely. We conclude with a qualitative discussion of the ideas behind the proof of the Geometrization Conjecture using the Ricci flow. Most of the project is based on the book by Chow and Knopf [6], the notes by Peter Topping [28] (which have recently been made into a book, see [29]), the papers of Richard Hamilton (in particular [9]) and the lecture course on Geometric Evolution Equations presented by Ben Andrews at the 2006 ICE-EM Graduate School held at the University of Queensland. -
About the Calabi-Yau Theorem and Its Applications
About the Calabi-Yau theorem and its applications { Krageomp program { Julien Keller L.A.T.P., Marseille University { Universit´ede Provence October 21, 2009 1 Any comments, remarks and suggestions are welcome. These notes are summing up the series of lectures given by the author during the Krageomp program (2009). Please feel free to contact the author at [email protected] 2 1 About complex and K¨ahlermanifolds In this lecture, starting with the background of Riemannian geometry, we introduce the notion of complex manifolds: • Definition using charts and holomorphic transition functions • Almost complex structure, complex structure and Newlander-Nirenberg theorem • Examples: the projective space , S2, hypersurfaces, weighted projective spaces • Uniformization theorem for Riemann surfaces, Chow's theorem (com- n plex compact analytic submanifolds of CP are actually algebraic vari- eties). We introduce the language of tensors on complex manifolds, and some natural invariants of the complex structure: • k-forms, exterior diffentiation operator, d = @ +@¯ decomposition, (p; q) forms • De Rham cohomology, Dolbeault cohomology, Hodge numbers, Euler characteristic invariant Using all the previous material, we can explain what is a K¨ahlermanifold. In particular we will formulate the Calabi conjecture on that space. • K¨ahlerforms, K¨ahlerclasses, K¨ahlercone n • The Fubini-Study metric on CP is a K¨ahlermetric Finally, we sketched the notion of holomorphic line bundle, the correspon- dence with divisor and Chern classes. This very classical material can be read in different books: - F. Zheng, Complex Differential Geometry, Studies in Adv. Maths, AMS (2000) 3 - W. Ballmann, Lectures on K¨ahlermanifolds, Lectures in Math and Physics, EMS (2006). -
Richard Bamler - Ricci Flow Lecture Notes
RICHARD BAMLER - RICCI FLOW LECTURE NOTES NOTES BY OTIS CHODOSH AND CHRISTOS MANTOULIDIS Contents 1. Introduction to Ricci flow 2 2. Short time existence 3 3. Distance distorion estimates 5 4. Uhlenbeck's trick 7 5. Evolution of curvatures through Uhlenbeck's trick 10 6. Global curvature maximum principles 12 7. Curvature estimates and long time existence 15 8. Vector bundle maximum principles 16 9. Curvature pinching and Hamilton's theorem 20 10. Hamilton-Ivey pinching 23 11. Ricci Flow in two dimensions 24 12. Radially symmetric flows in three dimensions 24 13. Geometric compactness 25 14. Parabolic, Li-Yau, and Hamilton's Harnack inequalities 29 15. Ricci solitons 32 16. Gradient shrinkers 34 17. F, W functionals 36 18. No local collapsing, I 38 n 19. Log-Sobolev inequality on R 39 20. Local monotonicity and L-geometry 40 21. No local collapsing, II 43 22. κ-solutions 44 22.1. Comparison geometry 45 22.2. Compactness 50 22.3. 2-d κ-solitons 50 22.4. Qualitative description of 3-d κ-solutions 51 References 53 Date: June 8, 2015. 1 2 NOTES BY OTIS CHODOSH AND CHRISTOS MANTOULIDIS We would like to thank Richard Bamler for an excellent class. Please be aware that the notes are a work in progress; it is likely that we have introduced numerous typos in our compilation process, and would appreciate it if these are brought to our attention. Thanks to James Dilts for pointing out an incorrect statement concerning curvature pinching. 1. Introduction to Ricci flow The history of Ricci flow can be divided into the "pre-Perelman" and the "post-Perelman" eras. -
C Omp Act Riemann
Jü rgen Jost C omp actRiemann S urfaces An Introduction to Contemporary Mathematics Third Edition With 23 Figures 123 Jü rgen Jost Max Planck Institute for Mathematics in the Sciences Inselstr. 22 04103 Leipzig Germany e-mail: [email protected] Mathematics Subject Classification (2000): 30F10, 30F45, 30F60, 58E20, 14H55 Library of Congress Control Number: 2006924561 ISBN-10 3-540-33065-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-33065-3 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Dupli- cation of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 PrintedinGermany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner, Heidelberg Typesetting by the author and SPI Publisher Services using a Springer LATEX macro package Printed on acid-free paper 11689881 41/sz - 5 4 3 2 1 0 Dedicated to the memory of my father Preface The present book started from a set of lecture notes for a course taught to stu- dents at an intermediate level in the German system (roughly corresponding to the beginning graduate student level in the US) in the winter term 86/87 in Bochum.