1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. SHRINKING BREATHERS ARE SHRINKING GRADIENT RICCI SOLITONS 245

(2) Furthermore, if A. (g (t)) > 0 and if v(g(t)) is not strictly increasing

on some interval, then g(t) is a gradient shrinking Ricci soliton.

(3) If v(g( to)) = -oo for some to, then v(g( t)) = -oo for all t E [O, to].

PROOF. (1) Given any 0 ~ti < t2 < T, we shall show that

(6.64). v(g(ti)) ~ v(g(t2)).

Since by assumption, v(g(t2)) > -oo, for any c: > 0 there exist f2 and T 2

such that

W(g(t2), h, T2) ~ v(g(t2)) + c:.

Let (f(t), T(t)), t E [O, t2], be a solution of the backward heat-type equation

(6.15) with f (t2) = f2 and T(t2) = T2· By the monotonicity formula (6.17),

we have^13

W(g(t2), f(t2), T (t2)) 2 W(g(ti), f(t1), T (t1)),

where equality holds if and only if

This implies

1

Rij + \li\ljf - -gij = 0 for all t E (t1, t2)·

2T

v(g(t2)) + c: 2 W(g(t2), f(t2), T (t2)) 2 W(g(ti), !(ti), T (t1)) 2 v(g(ti)).

The result follows since c: > 0 is arbitrary.

(2) Suppose v(g(ti)) = v(g(t2)) for some ti < t2. Since A. (g (t)) > 0, by

Corollary 6.34, there exist f2 and T2 such that

W(g(t2), f2, T2) = v(g(t2)).

In this case, by repeating the argument in (1), we obtain

W (g (t), f (t), T (t)) = v(g(t)) = const

for all t E [ti, t2]. As in the proof of Theorem 6.29, we can conclude that

g(t) ts a gradient shrinking Ricci soliton.

(3) If v(g(to)) = -oo, then for any N > -oo there exist Jo and To such

that W (g (to), Jo, To) ~ N. Let (f(t), T(t)), t E [O, to], be the solution of

(6.15) with f(to) = fo and T(to) =To. For all t E [O, to],

v(g(t)) ~ W (g (t), f (t), T (t)) ~ W(g(to), f(to), T (to)) ~ N.

Since N > -oo is arbitrary, we conclude v(g(t)) = -oo for all t E [O, to]. D

Using the v-invariant instead of the μ-invariant, we can give a

SECOND PROOF OF THEOREM 6.29. As we stated at the beginning of
this section, we only need to consider a shrinking breather g(t) with g(t 1 ) =

a*g(t2) and A. (g (t)) > 0, t E [ti, t2] · From the elementary properties (iii)