1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

72 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

We shall find it useful to consider solutions (including the King- Rosenau solu-

tion) in these various coordinate systems.

Let g(t), t E (-oo,O), be a maximal solution to the Ricci fl.ow on 52 , so that

the singularity time is 0. By applying the weak and strong maximum principles to

the equation for ~~, any compact ancient solution must h ave scala r curvature R

either zero everywhere or positive everywhere. Since we are on 52 , we must have

R > 0. By Hamilton [137], g (t) must shrink to a round point as t -t 0.

We may write

(29.4)

1 1 1

g(t) = V (t)g52 = V (t)9cyl = V (t)9euc,

where v ( t) is defined on 52 , v ( t) is defined on 51 x ~, and v ( t) is defined on ~^2.

For g(t) ~ v(t)9cyl and g(t) ~ v(t)9euc we have suppressed our need in (29.4) to

pull-back by the Mercator projection and stereographic projection, respectively.

The conformal factors u, u, and u are defined by

g(t) = U (t) g52 = U (t) 9 cyl = U (t) 9euc·

The Ricci fl.ow is equivalent to (the limiting case of) t he porous medium equation

ou

ot = 6.euc ln u.

In the study of the porous medium equation, the function v = i, is called the

pressure function. In the analysis of ancient solutions on 52 , it seems most

natural to consider v and its covariant derivatives with respect to g 52 as well. We

shall also call v and v pressure functions.

From the above discussion it is not difficult to see t hat we have the following

relations relating eit her the pressure functions or the conformal factors in various

coordinates:

(29.5)

and

(29.6)

v( s, e) t) = cosh^2 s v ('If; ( s)) e) t) , i.e., v( t) = cosh^2 s v( t) 0 m -l,

4u o 0"-^1

u = (1 + r2)2,

The scalar curvature of g (t) and the evolution of v (t) may be expressed as

(29. 7) O<R 9 =~av =~av =~av

vat vat vat

(^2) IV' l^2
- w.52V A - ---IV'vls2 + 2 V - t..i.A cy[V , - v , 9cy 1
v v
lv-1
2
- W.A eucV -- V 9 e u c ·
v
A significant part of this chapter is devoted to understanding the consequences of
this equation in the three different coordinate systems.
A fact that we sh all use is that since ft Area(g (t)) = - J 52 Rdμ, by the Gauss-
Bonnet formula and the extinction being at time 0 we h ave that the area of g ( t)
is 87r ltl.