1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

242 6. ENTROPY AND NO LOCAL COLLAPSING

3. Shrinking breathers are shrinking gradient Ricci solitons

Let (Mn,g(t)), t E [O,T), be a shrinking Ricci breather on a closed

manifold with g(t 2 ) = a<I>*g(t1), where t2 >ti and a E (0, 1). As discussed

at the beginning of Section 1 of this chapter, we only need to show that when

>..(g(t)) > 0 for all t E [t 1 , t 2 ], the shrinking breather is a shrinking Ricci

soliton. By Koiso's examples (subsection 7.2 of Chapter 2), such solutions

need not be Einstein.

In this section we give two variations on the proof that shrinking breathers

on closed manifolds are shrinking gradient solitons. The first proof, which

also appears in Hsu [208], involves fewer technicalities in that it uses μ in-

stead of v. The first proof also does not use the assumption>.. (g(t)) > 0. On

the other hand, the second proof requires some knowledge of the asymptotic

behavior of μ.

3.1. First proof using functional μ. The following result of Perel-

man rules out periodic orbits for the Ricci flow in the space of metrics

modulo diffeomorphisms and scalings.

THEOREM 6.29 (Shrinking breathers are gradient solitons). A shrinking

breather for the Ricci flow on a closed manifold must be a gradient shrinking

Ricci soliton.

FIRST PROOF. Let (M, g(t)) be a shrinking Ricci breather with g(t2) =

a<I>*g(ti), where t2 >ti and a E (0, 1). Define

so that ~; = -1,

T (t) =. t2 1 - at1 _ t '

-a

t2 -ti

T (t2) =Cl! l -a ,

and T (t2) = ar (ti). By Lemma 6.24, there is a minimizer fz for

{W(g (t2), f, T (t2)) : f E C^00 (M) satisfies (g, f, r) EX},

so that W(g(t2),Jz,r(t2)) = μ(g(t2),r(t2)). Define f(t) to solve (6.15)

on [t1, t2] with f (t2) = fz. By the monotonicity formula (6.17) and the
definition of μ, we have

μ (g (ti), T (ti)) SW (g (t1), f (ti), T (t1)) SW (g (t), f (t), T (t))

SW (g (t2), Jz, T (t2)) = μ (g (t2), T (t2))

for all t E [t1, t2] · Since g(t1) = a*g(t2) and T (t 2 ) = ar (t1), by the
diffeomorphism and scale invariance of μ, we have