1547671870-The_Ricci_Flow__Chow

32 2. SPECIAL AND LIMIT SOLUTIONS

In the formal limit m = 0, this equation may be applied to model the

thickness u > 0 of a thin lubricating film if one neglects certain fourth-order
effects. (See [19].) Since

um-1

lim = logu

m\.O m

for u > 0, the calculation

lim .6 (um -^1 ) = lim [um-^1 .6u + (m - 1) um-^2 [\7u[^2 ]

m\.O m m\.O

= .6u - [\7u[2 = .6logu

u u^2

shows that (2.20) is exactly the limit obtained for m = 0. This connection

between the porous media fl.ow and the Ricci fl.ow in dimension n = 2 was

first made by Sigurd Angenent. (See [128, 129].)

Now let h be the fl.at metric on the manifold M^2 = JR.xS[, where S[ is

the circle of radius 1. Give M^2 coordinates x E JR and () E S[ = JR./27rZ.

The Rosenau solution [111] of the Ricci fl.ow is the metric g = u·h defined

fort< 0 by

(2.22)

2/3 sinh ( -a>.t)

u (x t) = ------

' cosh ax + cosh a>.t

for parameters where a, /3,).. > 0 to be determined. Because u is independent

of (), we have

32

.6h log u = ox2 log u.

Hence the computations

( 2 _ 23 ) ~u (x t) = _ 2 a(3>. cosha>.t · coshax + 1

at ' (coshax + cosha>.t)^2

and

(2.24)^8

2

• 1 ogu ( x t ) =-a^2 cosha>.t · coshax +^1
  ox^2 ' (coshax + cosha>.t)^2
  show that u satisfies (2.20) (hence that g =uh solves the Ricci fl.ow on M^2 )
  if and only if

(2.25) 2(3).. =a.

Note that the Rosenau solution is ancient but not eternal, since by equation

(2.22),

lim u ( x, t) = 0.

t/O

By (2.19), the scalar curvature of g is given by

( 2. 26 ) R [g (t)] = _ .6h log u = a

2

cosh a>.t · cosh ax+ 1

u 2/3 sinh ( -a>.t) ( cosh ax + cosh a>.t) ·