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)