Math 241A -- Topics in geometry
Comparison theorems for Ricci curvature

Course Outline and Information

Fall 2003

Lecture: MWF 11:00 - 11:50; Girv 1112

Text: ``Math 241 Notes: 2000-2001" taken and typed by John Ennis. You can down load at http://www.math.ucsb.edu/ jennis/gradmath.html

 Instructor: Guofang Wei, South Hall 6503, Ext: 4282 


email: wei@math.ucsb.edu 

Office hours: MWF 9-10:00am or by appointment

Homework: There will be about three homework assignments, which will also be posted on my web page http://www.math.ucsb.edu/$\sim$wei.

Course outline:

We will study relations between curvature and topology. We will start by introducing the principal tool: comparison theorems for Ricci curvature, deriving relative volume comparison, Laplacian comparion (mean curvature comparison) and the segment inequality of Cheeger-Colding. From volume comparison we will prove Cheng's diameter sphere theorem, Milnor's growth of fundamental group, Gromov's bound and optimal bound of the first Betti number for almost nonnegative Ricci curvature, Anderson's theorem on the finiteness of isomorphism class of the fundamental groups. Using Laplacian comparison and maximal principle, we will prove other important tools like the splitting theorem, gradient estimate.

(Detailed outline can be found on the other side (or below))

References:  S. Zhu, The comparison geometry of Ricci curvature. Comparison 


geometry (Berkeley, CA, 1993-94), 221-262, Math. Sci. Res. Inst. Publ., 


30, Cambridge Univ. Press, Cambridge, 1997. 


J. Cheeger, Degeneration of Riemannian metrics under Ricci curvature 

bounds, Lezioni Fermiane, Pisa 2001.  


P. Petersen, Riemannian Geometry, GTM 171. Springer, 1998.  

(Chapters 9) 

Detailed outline:

   Part I: Volume Comparison Theorem 


1. What's the volume of a Riemannian manifold? 


2. How to calculate  the volume of a Riemannian manifold? 


3. Volume density comparison 


4. Bishop-Gromov's volume compariosm theorem 


5. The segment inequality of Cheeger-Colding  

   Part II: Applications of Volume Comparison Theorem


1. Cheng's maximal diameter theorem 


2. Growth of fundamental groups (Milnor's results) 


3. The first Betti number estimate 


4. Finiteness of fundamental groups (Anderson's result) 

   Part III: Laplacian comparison and its application  


1. What's the Laplacian?  


2. Laplacian comparison (mean curvature comparison) 


3. Maximal principle 


4. Splitting Theorem (by Cheeger-Gromoll) 


5. Applications of splitting theorem 


6. The gradient estimate  

Math 241B: Topics in geometry -- Ricci curvature and Gromov-Hausdorff Convergence

We will study Gromov-Hausdorff limits of manifolds with Ricci curavture bounded from below. We will start by introducing Gromov-Hausdorff convergence, proving Gromov's precompactness, then study the properties of the limit spaces. The goal is to introduce the major breakthrough of Cheeger-Colding in this direction, proving the regularity of the limit space. In particular we will show that volume is continuous under Gromov-Hausdorff convergence for mainfolds with Ricci curvature lower bounds.