1547671870-The_Ricci_Flow__Chow

LOCAL PINCHING ESTIMATES such that a solution exists, because d dt (>. - μ) = (>. - μ) (>. + μ - v) d dt (μ - v) = (μ ...

190 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE Let >. (lfD) 2: μ (lfD) 2: v (lfD) denote the eigenvalues of lfD E (A^2 T ...

LOCAL PINCHING ESTIMATES 191 0 0 where as in (6.34), Rm and Re are the trace-free parts of the Riemann curvature operator and R ...

192 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE where P may be written in terms of the eigenvalues >.., μ, v of Rm as (6.3 ...

LOCAL PINCHING ESTIMATES 193 Since the scalar curvature is R =tr (Rm) =tr (Re) = >.. + μ + v, it follows from (6.33) that ( ...

194 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE The result follows when we note that 1Rc l^2 :::; 1Rml^2 , discard the negati ...

THE GRADIENT ESTIMATE FOR THE SCALAR CURVATURE 195 To prove this estimate, we need to compute several evolution equations, the ...

196 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE The lemma follows by completing the square. 0 The evolution equation for f\7J ...

THE GRADIENT ESTIMATE FOR THE SCALAR CURVATURE 197 The useful term in the evolution equation for 1Rcl^2 - i R^2 is the quantit ...

198 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE one concludes that JV Rc\2 = \A\2 + \B\2 2 270 JV R\2' whence Hamilton's esti ...

THE GRADIENT ESTIMATE FOR THE SCALAR CURVATURE 199 The only potentially positive term in the evolution equation for V is 26 W ...

200 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE where C and I == 5 /2 depend only on go. On the other hand, it follows from f ...

7. HIGHER DERIVATIVE ESTIMATES AND LONG-TIME EXISTENCE 201 THEOREM 6.45. If go is a smooth metric on a compact manifold Mn, the ...

202 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE can accomplish this by the following strategy: when we have an upper bound on ...

7. HIGHER DERIVATIVE ESTIMATES AND LONG-TIME EXISTENCE 203 PROPOSITION 6.48. Let (Mn,g(t)) be a solution of the Ricci flow on a ...

204 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE Since Mn is compact, Jt I^8 Z;J (x, T) I dT .. 0 as t / T uniformly in g(t) x ...

HIGHER DERIVATIVE ESTIMATES AND LONG-TIME EXISTENCE 205 the metrics g (t) on the time interval [O, T). To estimate the first d ...

206 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE where the constants Ci, c~ depend only on m and n. For instance, the case m = ...

HIGHER DERIVATIVE ESTIMATES AND LONG-TIME EXISTENCE 207 Reasoning as in the derivation of (6.51) with the tensor Re replaced b ...

208 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE 7.2. Long-time existence. Now we are ready to show that the cur- vature becom ...