MATH 667- Modern Geometry-Fall 2012
Topic: Introduction to Ricci Flow
References: 1- Ben Andrews, Christopher Hopper, The Ricci Flow in Riemannian Geometry (Springer
2011)
2- Ben Chow, Dan Knopf, The Ricci
flow: an introduction (AMS 2004)
3- Ben Chow, Peng Lu, Lei Ni, Hamilton's
Ricci flow (AMS 2006)
4-Peter Topping, Lectures on Ricci
Flow (Cambridge 2006)
COURSE LOG
Th 8/23 Introduction, gradient flow of energy
(for curves)
Tu 8/28 gradient of energy for curves
Th 8/30 tension field, harmonic maps and flows
Tu 9/4, Th 9/6: harmonic map flow: subconvergence to harmonic map in
non-positive curvature
Tu 9/11: Ricci operator/linearization in local coordinates/ Moser
iteration (Brian)
Th 9/13: lack of ellipticity/ De Turck's existence scheme ([1, Ch.5])
Tu 9/18: Ricci-de Turck flow: existence ([1, Ch.5])
Th 9/20: Ricci-de Turck flow and the harmonic map flow: uniqueness/
evolution of scalar curvature
(Reference: [2, Chapter 3], [1, Ch.5])
Tu 9/25: A scalar comparison principle: applications to estimates for
R, vol, |Rm|^2
(Reference: [4, Chapter 3]
Th 9/27 Scaling estimates for space & time derivatives of Rm;
continuation criterion
(Reference: [1, Chapter 8]
Tu 10/2 Uhlenbeck trick: two versions [1, Ch. 6]/ Variation of
Einstein-Hilbert [1, Ch.10]
Th 10/4 Perelman's F-functional: first variation, associated system/
Maximum principle for symmetric 2-tensors
Planned presentations:
1- Perelman's F-functional and Ricci flow [1, 10.4], [4,
6.3/6.4] (Caleb)
2- Spacetime derivation of curvature evolution [1, 6.3.2] (Kevin)
3- Vector bundle maximum principle [1, 7.4] and [4, 9.6] (Einsof)
Tu 10/9 Preservation of
Ric>0/ Einstein tensor: evolution, diagonalization [4, Ch. 9]
Th 10/11 Fall Break
Tu 10/16 F-functional and monotonicity (Caleb) [1, 10.4]
Th 10/18 Vector bundle ODE/PDE principle (Einsof)