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{\bf Math 241A --- Topics in geometry} \\
Comparison theorems for Ricci curvature
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Course Outline and Information
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\hfill Fall 2003 

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\noindent{\bf Lecture}: MWF 11:00 - 11:50;  Girv 1112

\noindent {\bf 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

\begin{tabbing} {\bf Instructor}: \= Guofang Wei, South Hall 6503, Ext: 4282 \\
\> email: wei@math.ucsb.edu 
\end{tabbing}

\noindent {\bf Office hours}:  MWF 9-10:00am or by appointment
\newline

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

{\bf 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))

\begin{tabbing}
\noindent {\bf References}: 
 \=S. Zhu, {\em 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, {\em Degeneration of Riemannian metrics under Ricci curvature} \\
\>{\em bounds}, Lezioni Fermiane, Pisa 2001.  \\
\> P. Petersen, Riemannian Geometry, GTM 171. Springer, 1998.  \\
 \> (Chapters 9) 
\end{tabbing}

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\noindent {\bf Detailed outline}:

\begin{tabbing} 
  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
  \end{tabbing}
  \begin{tabbing} 
  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)
 \end{tabbing}
  \begin{tabbing} 
  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 
 \end{tabbing}
  \newline


 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.         
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