- ESTIMATES FOR POTENTIAL FUNCTIONS OF GRADIENT SOLITONS 11
PROOF. Let x EM - B 0 (2) be any point and let 1: [O, r (x)] --+ M be a unit
speed minimal geodesic joining 6 to x, where r(x) ~ d(x, 0). Define (: [O, r (x)]--+
[O, 1] to be the piecewise linear function(27.48) ((s)={ ~
r (x) - sif s E [O, 1],
ifs E (1,r(x)-1],ifs E (r(x)-1,r(x)].
Let {E1,... ,En-1,1'(0)} be an orthonormal basis ofT 0 M. Define Ei(s) E T-y(s)M
to be the parallel translation of Ei = Ei ( 0) a long r- Then the frame { E 1 ( s), ... ,
En-i(s),1^1 (s)} forms an orthonormal basis of T-y(s)M for s E [O, r (x)]. Since 1 is
minimal, by the second variation of arc length formula, we have for each ir(x)
0:::; O~Ei L(1) =lo (((')^2 - (^2 (Rm (1
1
, Ei) Ei, 1
1
)) ds.Summing over i, we obtain (compare with (27.18))(27.49)
r<x) r (x)
lo (^2 Re (1
1
, 1
1) ds:::; (n -1) lo ((')
2ds = 2(n - 1).
By applying the shrinker equation to Re and integrating by parts, we obtain(27.50) 2(n-1)?:: lo rr(x) (^2 (^1 )
2 - (f 01)" ds
1 2 11 1 r(x)
= -r(x) - - + 2 s (f o 1)
1
ds - 2 ( (f o 1)
1
ds
2 3 o r(x)-1
1 2 (1
?:: ( s)
2
r(x) -
3- 2 lo s Vf(O) +
2
ds-21r(x) ((s) (VJ(x)+ c;s)) ds,
r(x)-1
where for the last inequality we used by (27.30) thatI (f o 1)
1
(s) I:::; IV Jl(r (s)):::; Vf(O) + ~for s E [O, 1] and I (f o 1)^1 (s) I :::; VJ (x) + (~s) for s E [r(x) - 1, r(x)]. Hence
1 4 ;r- Ir
2(n-1)?:: 2r(x)- 3 - y f(O)-v f (x).That is ,(27.51)
Ir ;r- 1 - 2
v J (x) + v j(O)?::
2
d(x, 0) - 2n +
3
.If O is a minimum point of f, then f(O) :::; ~ by (27.43) and hence we have
VJ(x)?:: ~r(x) - i~n. D
REMARK 27.12. Compare (27.51) with Lemma 19.46 in Part III of Volume
Two.
COROLLARY 27 .13. For a complete normalized shrinker, if f attains its mini-
mum at 01 and 02, then(27.52)