RICCI SUMMER SCHOOL COURSE PLANS AND BACKGROUND READING LISTS

Transcription

1 RICCI SUMMER SCHOOL COURSE PLANS AND BACKGROUND READING LISTS The Summer School consists of four courses. Each course is made up of four 1-hour lectures. The titles and provisional outlines are provided below. For further information on the timetable and arrangements please refer to the Summer School webpage at Basic courses Course 1 Christina Sormani Comparison geometry with Ricci Bounds Prerequisite material Riemannian Manifolds, Geodesics and Exponential maps, Sectional Curvature, Gradients and Laplacians on Riemannian manifolds, (c.f. do Carmo s Riemannian Geometry Chapters 1-7), and the Maximum Principle. Course materials (all available online free) Uwe Abresch and Detlef Gromoll On Complete manifolds with nonnegative Ricci curvature" J. Amer. Math. Soc. 3 (1990), available free at Dimitri Burago, Yuri Burago and Sergei Ivanov A Course in Metric Geometry available at Peter Li Lecture Notes on Geometric Analysis available at Zhongmin Shen, Christina Sormani The Topology of Open Manifolds of Nonnegative Ricci Curvature arxiv: Christina Sormani, How Riemannian Manifolds Converge arxiv: The Definition of Ricci Curvature [Li, Chapter 1] The Bishop-Gromov Volume Comparison Theorem [Li, Chapter 2]

2 The Bocher-Weitzenboch formula [Li, Chapter 3] Gromov-Hausdorff Convergence [So, Section 3] [BBI, Chapter 7.3] Gromov's Compactness Theorem [So, Section 3] [BBI, Chapter 7.4] The Laplace Comparison Theorem [Li, Chapter 4] The Cheeger-Gromoll Splitting Theorem [Li, Chapter 4] The Abresch-Gromoll Excess Theorem [AbGr, Section 2]] Milnor s Conjecture on the Fundamental Group [ShSo, Section 3] Course 2 Joel Fine Introduction to Kaehler Geometry Definition(s) and examples of Kähler manifolds. First Chern class via Chern-Weil. Prescribing curvature of a line bundle and the d-dbar lemma. Positively curved line bundles. The Hörmander technique and peaked sections of positive line bundles. Kodaira embedding. Prescribing the volume form and finding Kahler-Einstein metrics via Monge-Ampère equation. A brief discussion of the analysis of the Monge-Ampère equation. Canonical Kahler metrics besides the Einstein case. The classical Futaki invariant. Reading D. Huybrechts. Complex Geometry: An Introduction. Springer, 2004 R. O. Wells Jr. Differential Analysis on Complex Manifolds. Springer, 2010 P. Griffiths and J. Haris. Principles of Algebraic Geometry. Wiley, 1978 G. Tian. Canonical Metrics in Kähler Geometry. Birkhaüser, 2000 Course 3 Brian Weber Introduction to epsilon-regularity and removable singularities theorems Prerequisites Basic Riemannian geometry and PDE. Suggested reading material J. Cheeger and D. Ebin, Comparison Theorems in Riemannian Geometry I. Chavel, Riemannian Geometry (Chapter VIII, Section IX.8)

3 A. Besse, Einstein Manifolds (Sections 0.E, 1.F; Chapters 4, 6, 12) D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order (Chapters 1-9, but particularly Chapter 8) Additional Reading Material M. Anderson, Orbifold compactness for spaces of Riemannian metrics and applications M. Anderson, Ricci curvature bounds and Einstein metrics on compact manifolds (1989) Bando, Kasue, Nakajima, On construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth (1989) C. Croke, Some isoperimetric inequalities and eigenvalue estimates (1980) Course Outline Lecture 1 - Manifolds and Analysis Introduction: The Search for Optimal Metrics and the Einstein-Hilbert functional Euler-Lagrange equations and elliptic system for curvature Sobolev and isoperimetric inequalities The significance of Ricci curvature in this context Lecture 2 - Geometric Convergence Review: Hausdorff and Gromov-Hausdorff convergence, and Gromov precompactness Cheeger Diffeofiniteness Lipschitz, C {k,a}, and L {k,p} convergence Integral and pointwise bounds on sectional curvature Lecture 3 - Epsilon-regularity and Orbifold Compactness Standard epsilon-regularity and epsilon-regularity on manifolds; the Nash-Moser process Curvature concentration and orbifold compactness C orbifold points Lecture 4 - Bubbling phenomena and other topics Bubbles, bubbles on bubbles, and bubble trees

4 Energy absorption and topological reduction Additional topics as time permits: moduli spaces of Einstein metrics, other canonical metrics, Kahler-Einstein manifolds, extremal Kahler metrics Intermediate courses Course 4 Hans-Joachim Hein Kähler-Einstein metrics on del Pezzo surfaces Background reading G. Tian, Kähler-Einstein metrics on algebraic manifolds, ICM lecture 1990 (free online). G. Tian, Existence of Einstein metrics on Fano manifolds, Cheeger Anniversary Volume, V. Tosatti, Kähler-Einstein metrics on Fano surfaces, arxiv: Preparatory Kähler manifolds with positive Ricci curvature [Ba Section 6] Aubin's continuity method role of the automorphism group [Ba Section 7, BM] del Pezzo surfaces and their automorphism groups largest time in Aubin s continuity method [Li, Sz] Tian's theorem, Step 1 [Di] the Aubin energies and the Ding Lagrangian properness of the Ding Lagrangian implies existence the -invariant; > dim/(dim + 1) implies properness Tian's theorem, Step 2,, and log-canonical thresholds [Ko, Shi] survey of computations of -invariants digression: other approaches to KE existence theorems [BiRo, Do]

5 if dim = 2 and then but [Shi] Tian's theorem, Step 3 a continuity method in the space of del Pezzo surfaces orbifold compactness for Einstein 4-manifolds uniform lower bounds on the Bergman kernel [DS, Ti Section 5] conclusion of proof using that An alternative ending equality of GH limits and algebraic limits [DS] classification of GH limits of KE del Pezzo surfaces [MaMu, OSS] References [Ba] W. Ballmann, Lectures on Kähler manifolds, EMS, Zürich, 2006 (free online). [BM] S. Bando, T. Mabuchi, Uniqueness of Einstein Kähler metrics modulo connected group actions, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987, [BiRo] O. Biquard, Y. Rollin, Smoothing singular extremal Kähler surfaces and minimal Lagrangians, arxiv: [Di] W. Ding, Remarks on the existence problem of positive Kähler-Einstein metrics, Math. Ann. 282 (1988), [Do] S. Donaldson, Kähler geometry on toric manifolds, and some other manifolds with large symmetry, arxiv: [DS] S. Donaldson, S. Sun, Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, arxiv: [Ko] J. Kollár, Which powers of holomorphic functions are integrable?, arxiv: [Li] C. Li, On the limit behavior of metrics in the continuity method for the Kähler-Einstein problem on a toric Fano manifold, Compos. Math. 148 (2012), [MaMu] T. Mabuchi, S. Mukai, Stability and Einstein-Kähler metric of a quartic del Pezzo surface, Lecture Notes in Pure and Appl. Math., 145, Dekker, New York, 1993, [OSS] Y. Odaka, C. Spotti, S. Sun, Compact moduli spaces of del Pezzo surfaces and Kähler-Einstein metrics, arxiv: [Shi] Y. Shi, On the alpha-invariants of cubic surfaces with Eckardt points, Adv. Math. 225 (2010), [Sz] G. Székelyhidi, Greatest lower bounds on the Ricci curvature of Fano manifolds, Compos. Math. 147 (2011), [Ti] G. Tian, On Calabi's conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990),

6 Course 5 Aaron Naber Cheeger-Colding theory: Structure and regularity of lowerricci curvature General references [Ch] Lecture Notes on Ricci Curvature, J. Cheeger [ChCo] On the Structure of Spaces with Ricci Curvature Bounded Below I & II, J. Cheeger and T. Colding. Lecture 1 - Harmonic Functions and Estimates In the first lecture we discuss the behavior of harmonic functions on manifolds with lower Ricci curvature bounds. We will discuss the Cheng-Yau gradient estimate as well as the use of distance functions to construct approximate Green s functions. Lecture 2 - Segment Inequalities and Almost Splitting Theorems The Splitting Theorem for manifolds M with Rc 0 says that if M contains a line then M = R N for some manifold N. An effective version of this statement is a key ingredient to the structure theory for Gromov Hausdorff limits of manifolds with lower Ricci curvature bounds. Namely, we will see that if M has Rc > ε and γ: [ ε 1, ε 1 ] is a long minimizing geodesic, then B 1 (γ(0)) is Gromov Hausdorff close to R N. Lecture 3 - Basic Structure Theory Using the almost splitting and almost volume cone theorems we will see how Gromov Hausdorff limits M i X may be stratified into a regular and singular part. We will discuss how in the non-collapsed case the singular part has codim at least 2, and the regular part is an actual topological manifold. Lecture 4 - Beyond Lower Ricci Curvature By combining the structure theory of the previous lectures with the standard epsilon-regularity theory of Anderson we see how non-collapsed GH limits of manifolds with bounded Ricci curvature are smooth away from a codim 2 singular set. In the Kahler case we discuss a refinement due to Cheeger and Tian which shows the singular set has codim 4.

7 Evening lecture abstracts Yanir Rubinstein Living on the edge, part I: analysis with metrics singular along a divisor; part II: who's afraid of polyhomogeneous expansions? Living on the edge can be tricky, even dangerous. We'll try to give a basic survival guide, based on our experience with Kahler edge metrics. Joel Fine The Kempf-Ness theorem and the intuitive picture behind correspondences of Hitchin-Kobayashi type I will outline the proof of the Kempf-Ness theorem which, put roughly, explains how to identify the symplectic quotient of a Kähler manifold X by the isometric action of a compact Lie group K and the complex quotient of X by the holomorphic action of the complexification G of K. The key is a certain geodesically convex function on the symmetric space G/K which enables one to relate stability to the zeros of the moment map. I will then describe how this result motivates conjectures in Kähler geometry, called "Hitchin-Kobayashi correspondences" which claim that certain stability conditions are equivalent to the existence of solutions to particular PDEs. One example is the equivalence of K-stability with the existence of a Kähler-Einstein metric, recently proved by Chen-Donaldson-Song and Tian.