1547671870-The_Ricci_Flow__Chow

3. ANCIENT SOLUTIONS 31

shows p (0) = 1 and p' (0) = 0 if and only if

lim 'ljJ (z) = 0

z-+w and

. w' (z)
Zlim -+W -(-) <{J Z = -1.

D

3.3. The Rosenau solution. In Lemma 5.3, we will show that if g
and h are metrics on a surface M^2 conformally related by g = ev h, then
the scalar curvature R 9 of g is related to the scalar curvature Rh of h by
R 9 = e-v (-~hv +Rh)· In the case that

g = U· h

for u : M^2 ---+ ffi+, this formula becomes

(2.19) Rg = -~h (logu) +Rh.

u

Now if u is allowed to depend on time but h is regarded as fixed, g will

satisfy the Ricci fl.ow equation for a surface,

if and only if

8

EJtg=-R·g,

8u. h = (~hlogu - Rh)

EJt u g,

hence if and only if u evolves by

EJu

EJt = ~h logu - Rh·

In the special case that ( M^2 , h) is fl.at, this reduces to

au I

(2.20)

8

= ~h l<Dg u.

REMARK 2.11. There is a: interesL ng connection between equation

(2.20) and the porous media !low, wht h is the PDE

(2.21) ~ u =~um.

UT I

When m = 1, this is just the ordinary linear heat equation. For all other

m > 0, it is nonlinear. When 0 < m < 1,lequation (2.21) appears in various

models in plasma physics. When m > I il, it models ionized gases at high

temperatures; and when m > 2, it models ideal gases fl.owing isentropically

in a homogeneous porous medium. (See [8].) If t ~ mT, then equation

(2.21) becomes I

~u= ~~um.

EJt m