1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED 203

On the other hand,

Plugging in the equation for (gt+~) Rm and using (5.40), we have

The last two terms cancel each other since \7 f = -\7 u / u, which yields the

lemma. D

REMARK 5.20 (Backward heat-type equation for modified scalar curva-

ture). From the proof of the lemma, we have

(5.44) .§_Rm= at -~Rm+ 2\7 Rm· \Jf + 2jR'f!l:ji1^2 ·

Note the similarity to the equation~~= ~R+2 jRcj^2 , except now we have

a backward heat-type equation.

3. Steady and expanding breather solutions revisited

A solution g(t) of the Ricci flow on a manifold Mn is called a Ricci

breather if there exist times ti < t2, a constant a > 0 and a diffeomorphism

<p : M --+ M such that

When a = 1, a < 1, or a > 1, we call g(t) a steady, shrinking, or

expanding Ricci breather, respectively. Recall that g(t) is a Ricci soliton
(or trivial Ricci breather) if for each pair of times ti < t 2 there exist a > 0
and a diffeomorphism <p : M --+ M (a and <p will in general depend on ti
and t2) such that g(t2) = a<p*g(ti).
Note that if we consider the Ricci flow as a dynamical system on the
space of Riemannian metrics modulo diffeomorphisms and homotheties, the
Ricci breathers correspond to the periodic orbits whereas the Ricci solitons
correspond to the fixed points. Since the Ricci flow is a heat-type equation,
we expect that there are no periodic orbits except fixed points.
A nice application of the energy monotonicity formula is the nonexis-
tence of nontrivial steady or expanding breather solutions on closed man-
ifolds (§2 of [297]). This was first proved by one of the authors in [218]
(see Proposition 1.66 in this volume). In the next chapter we shall see the
application of Perelman's entropy formula to prove shrinking breather solu-
tions on closed manifolds are gradient Ricci solitons (§3 of [297]). Hence we
confirm the above expectation.