1547671870-The_Ricci_Flow__Chow

NECKLIKE POINTS IN FINITE-TIME SINGULARITIES 269 subsequence of points {yi} in M^3 , orthonormal frames {Fi} at (Yi, ti), and ...

270 9. TYPE I SINGULARITIES We claim there is a constant r < oo depending only on g (0) such that A -----<r. A+μ+ 11 + p - ...

NECKLIKE POINTS IN ANCIENT SOLUTIONS 271 On the other hand, if .A (T - t) 2: c, then μ^2 + v^2 > 8.\^2 and we have d c 1 - ...

272 9. TYPE I SINGULARITIES PROOF. We may assume that supxEMn IRm(x,t)I is finite fort::; 0. Because Re 2': 0, we can estimate t ...

NECKLIKE POINTS IN ANCIENT SOLUTIONS 273 (When 'Y = 1, K corresponds to the bound in the definition of ancient Type I singular ...

274 9. TYPE I SINGULARITIES If (9.12) holds, then by Lemma 9.12 there is rt (6) such that 0 P 2: rt 1Rml^2 1Rml^2 , whence it fo ...

6. TYPE I ANCIENT SOLUTIONS ON SURFACES 275 whenever the solution exists for t E [-a, w). When the solution is ancient, letting ...

276 9. TYPE I SINGULARITIES whence it follows that the limit (9.13) E_ 00 ~ t->-00 lim E (h (t)) exists and is finite. We wan ...

NOTES AND COMMENTARY 277 Hence by the Compactness Theorem (Section 3 of Chapter 7), there is a subsequence (N^2 , hi ( t) , Xi) ...

278 9. TYPE I SINGULARITIES 9.9 is essentially Theorem 24.7 of [63]; the minor extension is that one as- sumes that the point is ...

APPENDIX A The Ricci calculus From the viewpoint of Riemannian geometry, the Ricci flow is a very natural evolution equation, be ...

280 A. THE RICCI CALCULUS Although there is no natural isomorphism (in the functorial sense) be- tween the bundles T Mn and T* M ...

2. FIRST-ORDER DIFFERENTIAL OPERATORS ON TENSORS 281 In terms of the Christoffel symbols (rt) induced by the Levi-Ci vita con- n ...

282 A. THE RICCI CALCULUS The Lie derivative is defined on vector fields so that .CxY= [X,Y] for all X, YE C^00 (T Mn), where [X ...

FIRST-ORDER DIFFERENTIAL OPERATORS ON FORMS 283 where { ei} ~=l is a (local) orthonormal frame field. If Mn is compact, the fo ...

284 A. THE RICCI CALCULUS for all e E OP (Mn) and 'f/ E OP+^1. By integrating by parts, it is easy to verify that n i=l for all ...

5. NOTATION FOR HIGHER DERIVATIVES 285 defined so that whenever (Mn, g) is Ricci-parallel - that is, whenever the (3, 0) -tensor ...

286 A. THE RICCI CALCULUS Commuting covariant derivatives Recall that the (3, 1)-Riemann curvature tensor is defined so that (A ...

APPENDIX B Some results In comparison geometry We begin this appendix by recalling basic results in comparison geometry that are ...

288 B. SOME RESULTS IN COMPARISON GEOMETRY Here vtv, Ww E Tw (TpMn) are the vectors identified with V, WE TpMn respectively, and ...