1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

210 5. ENERGY, MONOTONICITY, AND BREATHERS

SOLUTION TO EXERCISE 5 .27. We compute^13

_:!__ >. (g ( t)) I ~ lim inf >. (g (to)) - >. (g (to - h))

dt_ t=to h-+O+ h

> 1

.. f F (g (to) , f o) - F (g (to - h) , f (to - h))
_ imm h-+0+ h ,

where Jo is the minimizer for F (g (to),·) and f (t) is the solution to (5.54).

On the other hand, we conclude by (5.41) that the last expression is equal

to 2 JM IRij + \7i\7jfol^2 e-fodμ 9 (to)·

3.3. There are no nontrivial steady breathers. As an application

of the monotonicity of the diffeomorphism-invariant functional >. we prove

the nonexistence of nontrivial steady breathers.

LEMMA 5.28 (No nontrivial steady breathers on closed manifolds). If

(Mn,g(t)) is a solution to the Ricci flow on a closed manifold such that

there exist t1 < tz with>. (g (t1)) = >. (g (t2)), then g (t) is a steady gradient

Ricci soliton, which must be Ricci fiat. In particular, a steady Ricci breather

on a closed manifold is Ricci fiat.

PROOF. Note that if g (t) is a steady Ricci breather with g(t2) = cp*g(t1)

for some t1 < tz and diffeomorphism cp: M __, M, then >.(g (t2)) = >.(g (t1)).

Hence we only need to prove the first part of the lemma.

Suppose that for a solution g (t) to the Ricci flow there exist times t1 <

tz such that >.(g (t2)) = >.(g (t1)). Let fz be the minimizer for :F at time

tz so that F (g (t2), fz) = >. (g (t2)). Take f (t) to be the solution to the

backward heat equation (5.35) on the time interval [t1, tz] with the initial

data f (t2) = fz. By the monotonicity formula (5.41) and the definition of>.

we have^14

>. (g (t1)) -5:. F (g (ti), f (t1)) -5:. F (g (t), f (t)) -5:. F (g (t2), fz) = >. (g (t2))

for all t E [t1, tz]. Since>. (g (t1)) = >. (g (t2)) and >. (g (t)) is monotone, we

have

F (g (t), f (t)) = >. (g (t)) = const

fort E [ti, tz]. Therefore the solution f (t) is the minimizer for F (g (t), ·)

and 1:tF (g (t), f (t)) = 0, so by (5.41) we have

JM IRij + \7i\7jfl^2 e-f dμ (t) = 0

for all t E [t1, tz]. Thus

(5.57) Rij + \7i\7jf = 0 fort E [t1, tz].

In particular, g ( t) is a steady gradient Ricci soliton flowing along \7 f ( t).^15

(^13) Here d1_ denotes the lim inf of backward difference quotients.
(^14) This is the same as (5.55).
(^15) See (1.9), where a gradient soliton is steady if r=: = 0.