1547671870-The_Ricci_Flow__Chow

110 5. THE RICCI FLOW ON SURFACES

PROOF. Leth be a fixed metric on M^2 and let g be conformally related

to h by g = euh. By Lemma 5.3, the scalar curvatures of g and hare related
by

R 9 = e-u (- /j.hu +Rh).

If og/ot = f g, then ou/ot = f, and differentiating the equation above with
respect to time yields

~R = - (~u) e- u (-/j.hu +Rh) - e- ufj.h (~u) = - f R - /j. f.

at^9 at at^9 9

0

COROLLARY 5.8. Under the normalized Ricci flow on a surface, we have

(5.3)

a

ot R = jj.R + R (R - r),

where the average scalar curvature r is constant in time.

This type of evolution is known as a reaction-diffusion equation.

The Laplacian term promotes diffusion of R , whereas the quadratic reaction

terms promote concentration of R. If the right-hand side contained only

the Laplacian term, the equation would be the heat equation, albeit with

respect to a time-dependent metric; and one would expect R to tend to a

constant as t--+ oo. On the other hand, if the right-hand side contained only

the R (R - r) term, the equation would be an ODE; and the solution would

blow up in finite time for any initial data satisfying R (0) > max {r, O}.

The answer to the question of how the scalar curvature evolves under the

normalized Ricci fl.ow depends on which term dominates. We shall later see

that it is the diffusion ter~ which determines the qualitative behavior of the

equation.

As we discussed in Chapter 4, one may ignoring the Laplacian term in

the PDE (5.3) in order to compare its solutions with those of the ODE

(5.4)

d

dt s = s ( s - r) , s (0) =so,

where the function s = s (t) plays the role of R. When r -=/= 0 and so -=/= 0,

the solution of this ODE is

r

s (t) =.

1 - ( 1 - :0 ) ert '

when r = 0, we have s (t) = so/ (1 - sot); and when so = 0, the solution is

s(t) = 0. It follows that whenever so> max{r,O}, there is T < oo given by

T = { -~ log ( 1 - r /so) > 0 if r -=/= 0

1 I so > o if r = o

such that

lim s (t) = oo.

t-+T