1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

358 7. THE REDUCED DISTANCE

PROOF. Suppose limr--->O ft~~ (0) = V; the hypothesis implies by Lemma

7.92 that there is some E > 0 and some small neighborhood Uv of VE TpM

such that the map

(£exp,id): Uv x (f~E,f+c)--->M x (f-E,f+c),

(£exp, id) (W, T) = (£ 7 exp (W), T)

is a local diffeomorphism. For each WE Uv and T* E (f-E, f+E), we claim

the curve £exp(W,T), TE [0,T*], is a minimal £-geodesic. Hence using the

local diffeomorphism property, there are an E1 > 0, a small neighborhood

Uq of q, and a family of minimal £-geodesics "lq,r* smoothly depending on

the endpoint (j E Uq and 7* E (f - E1, f + E1). Now L ((j, T*) = £ ("fq_,rJ is a

smooth function of (q, T*), Lis differentiable near (q, f).

We now prove the claim by contradiction. If the claim is false, then there

is a sequence of points (Wi, Ti) ---> (V, f) such that £ exp(Wi, T), T E [O, Ti],

is not a minimal £-geodesic. Let £exp(Wi, T), T E [O, Ti], be a minimal

£-geodesic from p to £exp(Wi, Ti)· As in the proof of Lemma 7.28, using

Lemma 7.13(ii) and the £-geodesic equation, it is easy to show that IWil

g(p,0)

is bounded. Hence there is a subsequence Wi ---> W 00 • If W 00 =/= V, then we

get two minimal £-geodesics "IV and "fw 00 joining (p, 0) and (q, f), which

contradicts the assumption of the lemma. If W 00 = V, then (£exp, id) can-

not be a local diffeomorphism since (£ exp(Wi, Ti), Ti) = (.c exp(Wi, Ti), Ti).

The claim is proved and the lemma is proved. D

As a simple consequence, we have

(7.136)

is a diffeomorphism. Note that M\£ Cut(p,o) (f) is open and dense in M.

Now we end this subsection by rewriting the formula for the £-index

form in terms of Hamilton's matrix Harnack quadratic. Using

d~ [Re (Y, W)] = (! Re) (Y, W) + (\7 x Re) (Y, W)

+Re (\7xY, W) +Re (Y, \7xW),

we may write (7.134) as

1

Tb d

£I(Y, W) = - Ta VT dT [Re (Y, W)] dT

(gr Re) (Y, W) + !\7y\7wR-(Re (Y), Re (W))

1

(^7) b + (R (Y, X) W, X) - (\7y Re) (W, X)
+~a, , VT - (\7w Re) (Y, X) + 2 (\7 x Re) (Y, W)

+ (Re (Y) + \7 x Y, Re (W) + \7 x W)

dT.