Math 277: Ricci flow

For lecture notes and recordings, please scroll down

This is the website for the online class on Ricci flow, which I will teach in the fall semester of 2020 (August 27-December 8).

Please email me to be added to the mailing list of this class.

You are welcome to attend the class even if you are not a student at UC Berkeley. I am happy to email you the Zoom ID.

You can also solve the puzzle at the end of this webpage if you are in a rush and need the Zoom ID.

Due to certain security settings, you can only open the Zoom ID after creating a free Zoom account.

Meeting Times

The class will meet online over Zoom on Tuesdays and Thursdays 14:10-15:30 Pacific time. Please be aware that there will be a switch from daylight savings time on November 1, which is different from the day on which the rest of the world switches.

Office Hours

Fridays, 14:10-15:00 or by appointment. (Please note that I have to give priority to UC Berkeley students.)

Prerequisites

You should have some basic idea of Riemannian geometry (smooth manifolds, Levi-Civita connection, curvature, geodesics) and possibly some background in PDE. I don't think that you need to know too much, as long as you are willing read up on certain topics here and there. Surprisingly little knowledge of PDE will be required; most analytical estimates will have a geometric interpretation and will be introduced in this class.

Practice Problems

Further practice problems can be found here.

Note that there is no formal homework requirement for this class. These practice problems are optional.

I will update this file frequently and add problems as they come to my mind. Sometimes, I may add a problem or make adjustments to a previous problem set. So please reload this link frequently. The number of problems for each class may also vary, as certain topics lend themselves better or worse to practice problems.

There are no official solutions to these problems. You are welcome to discuss these problems with your classmates and I am also happy to set up a Piazza or Slack site for this if there is interest.

Tentative Schedule

I plan to spend a large part of the class discussing my recent work on the compactness and partial regularity theory of Ricci flows. Of course, I will also cover basic material in the beginning.

The following schedule may be too ambitious. I may need to adjust the schedule after getting some experience with the class. I may also decide to spend some more time on the 3-dimensional case (\(\kappa\)-solutions etc.) in the beginning of the class.

1 Th, 8/27 Basics in Riemannian Geometry [Pet, Lee, dCa], Basic properties of the Ricci flow, short-time existence [BK17, Appendix A], [Kry]
2 Tu, 9/1 Short-time Existence, Evolution of geometric quantities, Distance distortion estimates
3 Th, 9/3 Uhlenbeck's trick, Bianchi identity for the gradient of the heat equation, Evolution of curvature quantities
4 Tu, 9/8 Scalar weak and strong maximum principle, Applications
5 Th, 9/10 Further applications of the scalar weak and strong maximum principles, Local and global derivative estimates of the curvature tensor (Shi's estimates), Long-time existence, Barrier and Viscosity sense
6 Tu, 9/15 Maximum Principles in vector bundles, Applications
7 Th, 9/17 Rigidity discussion of the strong maximum principle, Hamilton's Ricci > 0 theorem
8 Tu, 9/22 Survey of other preserved curvature conditions, Hamilton-Ivey Pinching [Bre10, Bre19, BS09, BW08, Wil13]
9 Th, 9/24 Generalizations of the weak maximum principle [Bam15, BCW, Bei19], Geometric compactness theorems [BBS]
10 Tu, 9/29 Compactness of RFs, Further examples of singularity formation, Solitons, Perelman's \(\mathcal{F}\) and \(\mathcal{W}\)-functional
11 Th, 10/1 Perelman's No Local Collapsing Theorem based on the \(\mathcal{W}\)-functional, Perelman's Harnack estimate, \(\mathcal{L}\)-geometry
12 Tu, 10/6 \(\mathcal{L}\)-geometry continued, alternate proof of the No Local Collapsing Theorem
13 Th, 10/8 Hein and Naber's Poincare and Log-Sobolev inequalities and heat kernel bounds, Hypercontractivity, Integral Gaussian heat kernel estimates, pointed Nash-entropy
14 Tu, 10/13 Gradient estimates for heat equations, More on the pointed Nash-entropy, Variance bounds, lower volume bounds
15 Th, 10/15 More on heat kernel bounds, Upper heat volume bounds, Upper volume bounds, \(P^*\)-parabolic-neighborhoods, \(\varepsilon\)-regularity theorem
Tu, 10/20 NO CLASS
16 Th, 10/22 Basics on metric measure spaces and the Wasserstein distance
17 Tu, 10/27 Metric flows, basic properties, continuity of time-slices
18 Th, 10/29 The space of metric flow pairs, \(\mathbb{F}\)-convergence, compactness
19 Tu, 11/3 Intrinsic metric flows, the regular part of a metric flow, smooth convergence
20 Th,11/5 partial regularity of the limit
21 Tu, 11/10 partial regularity of the limit
22 Th, 11/12 partial regularity of the limit
23 Tu, 11/17 partial regularity of the limit
24 Th, 11/19 Ricci flows in dimension 3
25 Tu, 11/24 Ricci flows with surgery in dimension 3
Th, 11/26 NO CLASS
26 Tu, 12/1 Ricci flows with surgery in dimension 3
27 Th, 12/3 singular Ricci flows in dimension 3
28 Tu, 12/8 singular Ricci flows in dimension 3

Recordings and Lecture Notes

Recordings and lecture notes are available here. Please let me know if this link doesn't work properly, or if the download takes too long.

See also the typeset lecture notes by Ao Sun

Literature

[Bam15] R. Bamler, Stability of symmetric spaces of noncompact type under Ricci flow, Geom. Funct. Anal. 25 (2015), no. 2, 342-416.
[BBS] D. Burago, Y. Burago, S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, 33. American Mathematical Society Background on Gromov-Hausdorff convergence, length spaces, etc.
[BCW] R. Bamler, E. Cabezas-Rivas, B. Wilking, The Ricci flow under almost non-negative curvature conditions, Invent. Math. 217 (2019), no. 1, 95-126
[Bei19] S. Beitz, Bianchi-convex sets and a new maximum principle for the Ricci flow
[Bre10] S. Brendle, Ricci flow and the sphere theorem, Graduate Studies in Mathematics, 111. American Mathematical Society, Providence, RI, 2010
[Bre19] S. Brendle, Ricci flow with surgery on manifolds with positive isotropic curvature, Ann. of Math. (2) 190 (2019), no. 2, 465–559
[BK17] R. Bamler, B. Kleiner, Uniqueness and stability of Ricci flow through singularities Appendix A contains a discussion of the Ricci-DeTurck flow. Note that due to technical reasons we consider a slightly more general setting in this paper. More specifically, we allow the Ricci flow equation to hold up to a small error. I recommend setting this error to zero for the purpose of this class. This will simplify some computations.
[BS09] S. Brendle, R. Schoen, Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc. 22 (2009), no. 1, 287–307
[BW08] C. Böhm, B. Wilking, Manifolds with positive curvature operators are space forms, Ann. of Math. (2) 167 (2008), no. 3, 1079-1097
[CM] O. Chodosh, C. Mantoulidis, Lecture notes These are lecture notes from a previous Ricci flow class at Stanford. Towards the end of this class, I plan to cover different topics this time and hopefully improve certain parts, but they might still be helpful.
[CK] B. Chow, D. Knopf, The Ricci flow: an introduction, Mathematical Surveys and Monographs, 110, AMS
[CLN] B. Chow, P. Lu, L. Ni Hamilton's Ricci flow, Graduate Studies in Mathematics, 77, AMS
[Cho1] B. Chow et al, The Ricci flow: techniques and applications. Part I. Geometric aspects, Mathematical Surveys and Monographs, 135, AMS
[Cho2] B. Chow et al, The Ricci flow: techniques and applications. Part II. Analytic aspects, Mathematical Surveys and Monographs, 144, AMS
[Cho3] B. Chow et al, The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects, Mathematical Surveys and Monographs, 163, AMS
[Cho4] B. Chow et al, The Ricci flow: techniques and applications. Part IV. Long-time solutions and related topics, Mathematical Surveys and Monographs, 206, AMS
[dCa] M. P. do Carmo, Riemannian geometry, Birkhäuser More condensed than [Pet], [Lee]
[Kry] N. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, 12
[Lee] J. Lee, Introduction to Riemannian manifolds, Graduate Texts in Mathematics, 176 Background on Riemannian geometry, fewer details but easier read than [Pet]
[Pet] P. Peterson, Riemannian geometry, Graduate Texts in Mathematics, 171 Background on Riemannian geometry and geometric compactness theorems
[Top] P. Topping, Lectures on the Ricci flow
[Wil13] B. Wilking, A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities, J. Reine Angew. Math. 679 (2013), 223–247

Puzzle for the Zoom ID

In order to protect this class from Zoom-bombers, I need to ask you to solve one of my favorite puzzles (© Arthur Engel). The Zoom ID can be obtained by multiplying the answer of this puzzle with 45878932671.

Most inhabitants of the island of Sikinia are shepherds. It has been a long tradition that a sheep on Sikinia is worth as many Kolotniks (the local currency) as there are sheep its flock. Once upon a time, a shepherd died and left his entire flock to his son and daughter. The siblings decided to sell the flock and distribute the money evenly. In order to speed up the process, they placed all Kolotniks on a table and then alternatingly drew 10 Kolotniks from the heap until there was no money left over. The son drew first and the daughter ended up drawing the remaining Kolotniks.
At the end, the daughter complained: "That's unfair! There weren't 10 Kolotniks left over in the last turn."
The son replied: "You are right. Why don't you take my pocket knife. Then we are even."

How many Kolotniks is a pocket knife on the island of Sikinia?