1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. EXISTENCE OF A MINIMIZER FOR THE ENTROPY 23

Now suppose that (Mn,g(r),f(r)), r E (O,oo), is an expanding gra-

dient soliton solution to the backward Ricci fl.ow (i.e., g (t) ~ g (-t) is a

shrinking gradient Ricci soliton) satisfying

1

Rij + 'l/Jjf - 27 9ij = 0.

By the proof of Theorem 6.29 in Part I, f ( r) is a minimizer for W(g ( r) , · , r).

Note that g (r) is isometric to rg (1), so that

μ (g ( T) , T) = μ (g ( 1) , 1)

for all r E (0, oo ).

In the special case of an Einstein solution, where

1

Rij -

27

9ij = 0,

we have that a minimizer for W (g ( r) , · , r) is a constant function.^7 On the

other hand, by Lemma 17. 22 ( 1), given g ( r), for f sufficiently small, any

minimizer f:r of W (g ( r) , · , f) is not constant.

PROBLEM 17.23. It would be interesting to understand the behavior of

minimizers in some special cases.

(1) For r > 0 what are the minimizers of W (gsn, ·, r) on the unit n-

sphere? Are they always radial functions about some point in 5n7

How do the cases r < 2 (n~l) and r > 2 (n~l) compare? (Note that

R 95 n = n (n -1), so that in the case T = 2 (n~l) we have a constant

minimizer.)

(2) For which manifolds can one find minimizers with nice properties

such as having a certain amount of symmetry? For example, one

may consider the minimizers on complex projective space CPn.

3. Existence of a minimizer for the entropy

In this section we discuss the proof of the existence of a minimizer for
W which supplements the proof of Lemma 6.24 in Part I; we adopt here
the notation used there. Included in our discussion is a proof of a strong
maximum principle for weak solutions. We also consider a lower bound for
the maximum value of the minimizer.

3.1. Proof of the existence of a minimizer for W.

The following result is due to Rothaus and we follow his paper (see §1
of [161]).

PROPOSITION 17.24 (Existence of a smooth minimizer for W). For any

metric g on a closed manifold Mn and for any r > 0, there exists a smooth

minimizer fr of W (g, ·,r) which satisfies (17.14).

(^7) Note that the scaler is related to the scalar curvature by r = zRn.
g(r)