1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. PERELMAN'S iv-SOLUTION ON THE n-SPHERE

From (19.15) and (19.16) we obtain

Wlin (f (t), t):::; ~ + C (n) + ln VolB (po, Jt) -n - ~ ln (47rt)

= ln Vol B (po, Vt) - - - -n n ln ( ) 47r + C -( ) n

( v-tt 2 2.

for all Po EM and t E (0, oo). Since Wlin (f, t) 2:: -C, we conclude that

(^1) n VolB ( Vlt (po, Vt) > -:-. -C _ 0 - ( ) n
for all Po EM and t E (0, oo ).
93

D

REMARK 19.35. Assuming maximum volume growth, a precise relation

between the asymptotic behaviors of volume growth and the linear entropy

of a heat kernel is given by the following (see Corollary 4.3 of [139])

inf Wlin (f (t), t) = lim Wlin (f (t), t) = ln (AVR(g)) :::; 0.

tE(O,oo) t-+oo

In the case of the (backward) Ricci flow g ( T) on a closed manifold and

the adjoint heat kernel u centered at (po, 0), recall from inequalities (16.106)

and (16.105) in Part II that

f (y, T) :::; £ (y, T) ,

where u = (47rr)-n/^2 e-f and £ is the reduced distance based at (po, 0).

Integrating

we see that

V(r):::; 1,

where V is the reduced volume (see the next chapter for a discussion of V).

Recall also that by Theorem 16.44 in Part II we have

W (g ( T) , U ( T) , T) :::; 0.

It is possible that lim 7 -+oo W (g ( T) , u ( T) , T) is equal to ln ( lim 7 -+oo V { T))

or that in some other way these two invariants are related?

Note that for solutions of the Ricci flow as compared to the static metric

case, from this perspective, the reduced volume seems like a natural replace-

ment for the asymptotic volume ratio (recall from Corollary 20.2 that any

ancient solution with bounded Rm;:=:: 0 must have AVR = 0).

3. Perelman's 11;-solution on then-sphere
   A higher-dimensional analogue of the 2-dimensional King-Rose:q.au solu-
   tion is Perelman's 11;-solution. This is the rotationally symmetric 11;-solution
   on S^3 constructed in §1.4 of [153]. It is possible, as with other ancient
   solutions, that Perelman's solution has physical significance.