 
Summary: MATH 241A Topics in Differential Geometry
Fall 2011, TR 2:003:15, SH 4519
Instructor: Prof. Rick Ye
Office: SH 6509, Tel. 8938034, email: yer@math.ucsb.edu
Office Hours: TR 12:451:55, SH6509.
Introduction to Einstein Manifolds and the Ricci Flow
In this course we'll introduce the concepts of Einstein manifolds and Ricci
flow and present some basic results in these two directions. The following is
a list of topics on the plan:
1. Basic properties of the Einstein equation. Gauge fixing and ellipticity.
2. The total scalar curvature functional and its relation to Einstein metrics.
Its first variation and second variation.
3. Decomposition of the Riemann curvature tensor.
4. Locally conformally flat manifolds. The conformal Laplacian. Introduc
tion to the Yamabe problem.
5. Obata's theorem.
6. The total scalar curvature functional on asymptotically flat manifolds.
7. Topology of Einstein 4manifolds.
8. Short time existence of the Ricci flow.
9. Perelman's entropy functional, the log entropy functional of Ye and non
