Comments on: 285G, Lecture 8: Ricci flow as a gradient flow, log-Sobolev inequalities, and Perelman entropy

By: 285G, Lecture 17: The structure of κ-solutions « What’s new

285G, Lecture 17: The structure of κ-solutions « What’s new — Tue, 03 Jun 2008 20:36:03 +0000

[...] shift time so that the volume is in fact equal to , and consider the Perelman entropy defined in Lecture 8. Testing this entropy with ) we obtain an upper bound . On the other hand, on the sequence of times [...]

By: 285G, Lecture 15: Geometric limits of Ricci flows, and asymptotic gradient shrinking solitons « What’s new

Wed, 28 May 2008 02:02:18 +0000

[...] 2. If (M,g) is Hamilton’s cigar (Example 3 from Lecture 8), and is a sequence on M tending to infinity, then converges geometrically to the pointed round [...]

By: 285G, Lecture 11: κ-noncollapsing via Perelman reduced volume « What’s new

Mon, 19 May 2008 18:00:44 +0000

[...] 2 from Lecture 7. Of course, we already proved (a stronger version) of this theorem already in Lecture 8, using the Perelman entropy, but this second proof is also important, because the reduced volume is [...]

By: 285G, Lecture 12: High curvature regions of Ricci flow and κ-solutions « What’s new

Sat, 17 May 2008 00:14:00 +0000

[...] 16 May, 2008 in 285G - poincare conjecture, math.DG by Terence Tao Tags: canonical neighbourhoods, compactness and contradiction, gradient shrinking solitons, kappa-solutions In previous lectures, we have established (modulo some technical details) two significant components of the proof of the Poincaré conjecture: finite time extinction of Ricci flow with surgery (Theorem 4 of Lecture 2), and a -noncollapsing of Ricci flows with surgery (which, except for the surgery part, is Theorem 2 of Lecture 7). Now we come to the heart of the entire argument: the topological and geometric control of the high curvature regions of a Ricci flow, which is absolutely essential in order for one to define surgery on these regions in order to move the flow past singularities. This control is intimately tied to the study of a special type of Ricci flow, the -solutions to the Ricci flow equation; we will be able to use compactness arguments (as well as the -noncollapsing results already obtained) to deduce control of high curvature regions of arbitrary Ricci flows from similar control of -solutions. A secondary compactness argument lets us obtain that control of -solutions from control of an even more special type of solution, the gradient shrinking solitons that we already encountered in Lecture 8. [...]

By: 285G, Lecture 10: Variation of L-geodesics, and monotonicity of Perelman reduced volume « What’s new

Sun, 11 May 2008 14:52:10 +0000

[...] that . Note that this fact implies the monotonicity of Perelman reduced volume (cf. Exercise 2 from Lecture 8). [It seems that the elliptic analogue of this fact is the assertion that the Newton-type potential [...]

By: 285G, Lecture 9: Comparison geometry, the high-dimensional limit, and Perelman reduced volume « What’s new

Mon, 28 Apr 2008 02:09:17 +0000

[...] the volume comparison result whenever one has bounded normalised curvature, which was used in the previous lecture; indeed, thanks to the above inequality, it suffices to prove the claim for model [...]

By: Terence Tao

Terence Tao — Sat, 26 Apr 2008 04:55:51 +0000

Thanks for the corrections!

By: Dan

Dan — Sat, 26 Apr 2008 02:50:38 +0000

A few little typos:

The equations are not uniquely numbered from 46 to 49. The exercises are not uniquely numbered from 9 to 10.

Some parentheses would be useful in equations 32 and (the second) 46.

In Exercise 5, Remark 1, and (the second) Exercise 9, I think you want to say “isometric” instead of “diffeomorphic.”

At the beginning of the “non-collapsing” section, there is a reference to equation 35 that doesn’t make sense. Perhaps you meant 45?

Finally, there is a typo in equation 53, and I also think that the sign on is wrong (and persists in later equations).

Thank you for the lucid discussion of the Perelman entropy (and related topics).

By: Américo Tavares

Américo Tavares — Fri, 25 Apr 2008 20:05:00 +0000

Minor typo: 285A, in the title, should be 285G