USTC-2015

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.

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

Lesson I: (July 7) 2-4 pm

• 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) 2-4 pm

• 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]