1547671870-The_Ricci_Flow__Chow

AN ALTERNATIVE STRATEGY FOR THE CASE x (M^2 > 0) 169 To prove the claim we consider the trace Harnack quantity for the un- ...

170 5. THE RICCI FLOW ON SURFACES PROPOSITION 5.97. Let (M^2 ,g (t)) be a solution of the Ricci flow on a topological 2-sphere w ...

NOTES AND COMMENTARY 171 by (5.52) and (5.53), we have R(-,[) =r(T-t) R(-,t). But since g (t) has a Type I singularity, the righ ...

172 5. THE RICCI FLOW ON SURFACES in [ 28 ]. A new proof without the use of the potential function was given by Hamilton and Yau ...

CHAPTER 6 Three-manifolds of positive Ricci curvature The topic of this chapter is Hamilton's application of the Ricci flow to t ...

174 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE of the unnormalized flow modulo rescaling; but the advantage of finally conve ...

THE EVOLUTION OF CURVATURE (5) The volume form dμ of g evolves by a 1 ~dμ = - (tr 9 h) dμ. ut 2 Substituting h = -2 Re into th ...

176 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE LEMMA 6.7. The scalar curvature of a solution to the Ricci flow evolves by (6 ...

THE EVOLUTION OF CURVATURE 177 This identity shows in particular that the Ricci tensor determines the Rie- mann tensor when n ...

178 6. THREE-MANIFOLDS OF POSITIVE RICCI C URVATURE PROOF. By applying the second Bianchi identity and commuting covari- ant der ...

THE EVOLUTION OF CURVATURE 179 COROLLARY 6.14. Under the Ricci fiow, the (4, 0)-Riemann curvature tensor satisfies the followi ...

180 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE Uhlenbeck's trick Let (Mn,g(t)) be a solution of the Ricci fl.ow, and let { ...

UHLENBECK'S TRICK 181 restrictions (lo)x : Vx ---+ TxMn are vector space isomorphisms depending smoothly on x E Mn.) Then if w ...

182 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE Using the usual product rule, \7 (t) and D (t) define connections on tensor p ...

THE STRUCTURE OF CURVATURE EVOLUTION 183 we compute 3. The structure of the curvature evolution equation By looking more close ...

184 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE whence (Rm (U))ij = "'(Uij - Uji) = 211,Uij· Now we can square the operator R ...

THE STRUCTURE OF CURVATURE EVOLUTION 185 We now adopt this general definition to introduce another square of the operator Rm. ...

186 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE Having completed these constructions, we can now write the evolution of the R ...

REDUCTION TO THE ASSOCIATED ODE SYSTEM 187 Having written the evolution of the curvature operator in this form, we can immedia ...

188 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE exists a globally defined orthonormal moving frame {ei}· We fix an or- thonor ...