54 28. SPECIAL ANCIENT SOLUTIONS
M x ( -oo, 0) -+ JR satisfies 6.f = R. Then, for any smooth bounded domain
n c M and any time t < 0, we have
where v is the unit outward normal to 80; all norms, covariant derivatives, and
Laplacians are with respect to g( t).PROOF. Fix t < 0. Recall the everywhere present Bochner formula
(28.44) 6. IV' fl^2 = 2 (V'(6.f), V' f) + 2 R e (V' f , V' f) + 2 IV'^2 fl
2= 2 (V'R, V'f) + R IY'fl^2 +2IV'^211
2
.From this we obtain(28.45)6.(R + IV' fl2) = 6.R + R2 - (6.!)2 - IV' Rl2
R
- JV' :1
2
+ 2 (V' R , V' f) +RIV' !1
2
+2IV'^2 f12= IY'R+RV'fl2 +21V'2f-~6.fgl2 +6.R+R2_ IY'Rl2.
R 2 R
Hamilton's trace Harnack estimate (see Proposition 15.7 in Part II) says(28.46) 8R at = 6.R+R2 > - IY'Rl2 R > - o.
Thus, by applying the divergence theorem to the integral of (28.45) over n, we
obtain (28.43). 0Modulo some estimates to be subsequently proved, we can now give thePROOF OF THEOREM 28.41. Fix any time t < 0. Choose fl = Oi in (28.43),
where Di is as in Lemma 28.45. By taking i-+ oo we obtain
Now, (28.51) and (28.70) below imply that the RHS of this tends to zero. We
conclude that V'^2 f = ~Rg =Re on M x (-oo, O); i.e., g (t) is a steady GRS flowing
along -V'f. By Proposition 1.25 in Part I , (M,g(t)) must be a constant multiple
of the cigar soliton. 0
We shall call Ian; v(R)dO" and Ian; v(IV' fl
2
)dO" on the RHS of (28.47) to be the
first and second boundary terms, respectively. In the following subsections, we
prove that both of these boundary terms tend to zero.