1547671870-The_Ricci_Flow__Chow

ANCIENT SOLUTIONS 29 FIGURE 1. Shrinking round n-sphere 3.2. The cylinder-to-sphere rule. Before considering our next ex- ampl ...

30 2. SPECIAL AND LIMIT SOLUTIONS and define 7/J(z ) p(r) = p(r(z)). ~ - r. Then the metric may be written as g = dr2 + r2p2 de2 ...

ANCIENT SOLUTIONS 31 shows p (0) = 1 and p' (0) = 0 if and only if lim 'ljJ (z) = 0 z-+w and . w' (z) Zlim -+W -(-) <{J Z = ...

32 2. SPECIAL AND LIMIT SOLUTIONS In the formal limit m = 0, this equation may be applied to model the thickness u > 0 of a t ...

ANCIENT SOLUTIONS 33 In particular, the solution has positive curvature for as long as it exists. Now regard M^2 as the 2-sphe ...

34 2. SPECIAL AND LIMIT SOLUTIONS Moreover, the curvature R±oo ( t) at the poles is actually the maximum cur- vature of (S^2 , g ...

IMMORTAL SOLUTIONS 35 Consider the one-parameter family of expanding metrics g ( t) defined for t > 0 in the polar coordina ...

36 2. SPECIAL AND LIMIT SOLUTIONS (Compare with the Ricci- DeTurck flow analyzed in Section 3 of Chapter 3.) If we can construct ...

IMMORTAL SOLUTIONS 37 Notice that f (r) ----? 1 is possible only if r ----? oo, so that we either have 0 < f (r) < 1 or ...

38 2. SPECIAL AND LIMIT SOLUTIONS \ \ I I I J I I I FIGURE 2. A neckpinch forming The neckpinch The shrinking sphere we conside ...

THE NECKPINCH 39 FIGURE 3. The shrinking cylinder soliton A neckpinch is a special type of local singularity. There exist quan ...

40 2. SPECIAL AND LIMIT SOLUTIONS anything we have presented thus far, but illustrate many important ideas that will be develope ...

THE NECKPINCH 41 REMARK 2.17. In the special case of a reflection-symmetric metric on sn+l with a single neck at x = 0 and two ...

42 2. SPECIAL AND LIMIT SOLUTIONS and its scalar curvature is (2.52) R = 2nKo + n (n - 1) K1. 5 .2. Bounds on curvature and othe ...

5. THE NECKPINCH 43 PROOF. Applying the maximum principle to (2.54), one concludes that at any maximum of v which exceeds 1, one ...

44 2. SPECIAL AND LIMIT SOLUTIONS COROLLARY 2.23. sup Ja (-, t) i :::::; a~ sup la(-, O)J. We can also show that the curvature i ...

THE NECKPINCH one calculates L SS = -2 'l/J'l/Jsss 1f;2 + 6 'l/J; 1f;2 (K + L) + 2 ('lf;; 1f;2 -L) K - 2K^2. Combining these e ...

46 2. SPECIAL AND LIMIT SOLUTIONS -1 x (t) * FIGURE 4. A typical profile PROOF. Noting that "'' _ a + 'l/J; - 1 'f/SS - 'l/J ' o ...

THE NECKPINCH 47 ( 2) There exists a time T bounded above by r min ( 0)^2 / ( n - 1) such that the radius rmin(t) of the small ...

48 2. SPECIAL AND LIMIT SOLUTIONS To prove the claim, fix some to such that 'ljJ (-,to) is a Morse function, and let its smalles ...