授 课 人：Prof. Christina Sormani

日期：7.7―7.24
时间：周二， 周四 14:00―16：00；7月23日（周四）下午：6：00-8：00

地点：管理楼1308

课程：Ricci Curvature and Convergence

Course Overview

We will cover Gromov-Hausdorff Convergence of Riemannian Manifolds and Metric Spaces, Cheeger-Colding Theory for Manifolds with Ricci Curvature bounded below and Intrinsic Flat Convergence of Oriented Riemannian manifolds and Integral Current Spaces.   These techniques have been
applied to understand singularities which develop under various flows and limits in Riemannian Geometry including work of Hamilton and Perelman on Ricci Flow, work of Chen-Donaldson-Sun and Tian on Kahler Einstein Manifolds, and the ongoing study of the stability of the Schoen-Yau/Witten
Positive Mass Theorem.   All course materials are available online and the course outline and detailed topics are below.

Course Materials 

[AG] Abresch-Gromoll “On Complete manifolds with nonnegative Ricci curvature"  J. Amer. Math. Soc. 3 (1990), 355-374.  http://projecteuclid.org/euclid.jdg/1214430220

[BBI] Burago-Burago-Ivanov “A Course in Metric Geometry”  (a textbook)  http://www.math.psu.edu/petrunin/papers/alexandrov/bbi.pdf

[ChCo] Cheeger-Colding "On the Structure of Manifolds with Ricci Bounded Below I" Annals 1996 https://projecteuclid.org/euclid.jdg/1214459974
see also mathscinet review by Minicozzi

[LS] Lee-Sormani "Stability of the Positive Mass Theorem for Rotationally Symmetric Riemannian Manifolds" Crelle 2014 http://arxiv.org/abs/1104.2657

[HLS] Huang-Lee-Sormani "Intrinsic Flat Stability of the Positive Mass Theorem for graphical hypersurfaces of Euclidean Space" http://arxiv.org/abs/1408.4319

[Li] Li, Peter "Lectures Notes on Geometric Analysis" http://www.docin.com/p-294646772.html  or  http://www.researchgate.net/publication/2634104_Lecture_Notes_On_Geometric_Analysis

[ShSo]  Shen-Sormani  “The Topology of Open Manifolds of Nonnegative Ricci Curvature” (a survey)  http://arxiv.org/abs/math/0606774

[S] Sormani, "Nonnegative Ricci curvature, Small Linear Diameter Growth and Finite Generation of Fundamental groups" JDG 54 (2000), no. 3, 547--559.  http://arxiv.org/abs/math/9809133

[So] Sormani, “How Riemannian Manifolds Converge” (a survey)  http://arxiv.org/abs/1006.0411

[Sor] Sormani, "Intrinsic Flat Arzela-Ascoli Theorems" http://arxiv.org/abs/1402.6066

[Sor2] Sormani, "Properties of Intrinsic Flat Convergence"

[SW] Sormani-Wenger "Intrinsic Flat Distance between Riemannian Manifolds and other Integral Current spaces" JDG 2011 http://arxiv.org/abs/1002.1073

[SoWei] Sormani-Wei "The Covering Spectrum..." http://arxiv.org/abs/math/0311398

[Wei] Wei, Guofang "Manifolds with a Lower Ricci Curvature Bound" (Survey 2006) http://mail.math.ucsb.edu/~wei/paper/06survey.pdf

Updated Course Outline (changed to match Prof Cai's course)

Lesson I: (July 7) Introduction and Metric Geometry Background
Introduction: How Riemannian Manifolds Converge [Figures in So 3]
Metric Spaces cf. [BBI 1.1] and Riemannian Manifolds as Metric Spaces cf. [BBI 5.1]
Compactness and Completeness cf. [BBI 1.5] [BBI 1.6]
Hausdorff Convergence of sets cf. [So 2.1]
Hausdorff Measure and Dimension cf. [BBI 1.7] [So 2.1]
Length Spaces cf. [BBI 2.1-2.5] and Riemannian Manifolds as Length Spaces cf. [BBI 5.1]
Lesson II: (July 9) Gromov-Hausdorff Convergence
Hausdorff Convergence cf. [So 2.1]
Gromov-Hausdorff Convergence cf. [So, 3.1] [BBI, Chapter 7.3]
Gromov's Compactness Theorem cf. [So, 3.1] [BBI, Chapter 7.4]
Gromov's Embedding Theorem  [Gromov "Groups of Polynomial Growth"]
Pointed Gromov-Hausdoff Convergence cf. [BBI 8.1]
Tangent and Asymptotic Cones cf. [BBI 8.2]

Lesson III: (14/07/2015)
Gromov-Hausdorff Bolzano-Weierstrass Thm [BBI]
Pointed Gromov-Hausdorff Convergence [BBI]
Gromov's Ricci Compactness Theorem cf. [Wei]
Colding's Volume Convergence cf. [Wei]
Cheeger-Colding Maximal Volume Theorem cf. [Wei]

Lesson IV: (16/07/2015)
Gromov-Hausdorff Arzela-Ascoli Theorem [BBI]
Topology of Gromov-Hausdorff Limits [SoWei]
Sormani-Wei Covering Spectrum Convergence [SoWei]
Topology of Limits of manifolds with Ricci /ge 0 [SoWei]
Almost Rigidity Theorems (Cheeger-Colding Almost Splitting Theorem) cf. [Wei]
Metric Measure Convergence (Fukaya) cf. [Wei]
Cheeger-Colding Structure of Limit Spaces cf. [Wei]

Lesson V: (21/07/2015)
Sormani-Wenger Intrinsic Flat Convergence [SW] cf. [So]
Federer Flemming Flat convergence of Currents in Euclidean Space cf [SW]
Ambrosio-Kirchheim Currents on Metric Spaces cf [SW]
Ambrosio-Kirchheim Compactness Theorem cf [SW]
Intrinsic Flat limits in Gromov-Hausdorff limits [SW]

Lesson VI: (23/07/2015)
Intrinsic Flat =GH convergence for Ricci nonnegative noncollapsing [SW2]
Cancellation under Intrinsic Flat Convergence [SW2]
Wenger's Compactness Theorem cf. [SW]
Almost Rigidity/Stability of Schoen Yau Positive Mass Theorem [LS][HLS]
Intrinsic Flat Arzela Ascoli Theorems [Sor]
Tetrahedral Compactness Theorem [Sor2]