1547671870-The_Ricci_Flow__Chow

5. LOCAL PINCHING ESTIMATES

such that a solution exists, because

d

dt (>. - μ) = (>. - μ) (>. + μ - v)

d

dt (μ - v) = (μ - v) ( ->. + μ + v).

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By (6.30), the eigenvalues >., μ, v of M are twice the sectional curvatures.
Hence at x E M^3 , we may regard the Ricci tensor as the matrix

(6.33) Re=-1 ( μ+v >.+v ).

2 >.+μ

For later reference, we note that the trace-free parts of Rm and Re are
related at the same point by

(6.34) R~ = ~ (

2

>. - μ - v ->. + 2μ - v ) = -2Rc.

->.-μ+ 2v

5. Local pinching estimates

Now we are ready to state the pinching results which are true for 3-

manifolds with positive Ricci curvature. The first estimate proves that cur-

vature pinching is preserved, whereas the second shows that it improves,

hence that a solution to the Ricci flow on a 3-manifold with positive Ricci

curvature is nearly Einstein at a ny point where its scalar curvature is large.

These are pointwise estimates; below, we shall develop techniques to com-

pare curvatures at different points of a solution.

As in Section 4, we let >. (t) 2:: μ (t) 2:: v (t) denote the eigenvalues of

the curvature operator Rm of a solution (M^3 , g (t)) to the Ricci flow on a

closed 3-manifold, recalling that these are twice the sectional curvatures.

LEMMA 6.28 (Ricci pinching is preserved.). Let (M^3 ,g(t)) be a solution

of the Ricci fiow on a closed 3-manifold such that the initial metric go has

strictly positive Ricci curvature. If there exists a constant C < oo such that

(6.35) >.:::; C(v+μ)

at t = 0, then this inequality persists as long as the solution exists.

PROOF. Recall that>. 2:: ~ (v + μ) > 0, so that we can take logarithms.

Compute

-log d ( -->. ) =^1 ( (v+μ)-->.d>. -d (v+μ) )

dt v + μ >. ( v + μ) dt dt

1

= >.(v+μ) ((v+μ)(>.2+μv)->.(v2+>.μ+μ2+>.v))

(6.36)

= μ2 (v - >.) + v2 (μ - >.) < 0.

>.(v+μ) -