1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

478 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS

THEOREM B.2 (Wu [373]). Let go be a complete metric on lR^2 , of the
form (B.1), with R and J'VuJ (measured using go) bounded, and JR_ dμ
finite. Then a solution to the Ricci flow exists for all time, the aperture
and circumference at infinity are preserved, and the metric has bounded

subsequence convergence as t -+ oo. If R(go) > 0 and A(go) > 0, then the

limit is fiat. If R(go) > 0 and C 00 (go) < oo, then the limit is the cigar.

Note that C 00 being finite implies that A= 0. However, there are plenty
of complete metrics with A = 0 and C 00 = oo, for which the limit of the
Ricci fl.ow is not classified. An example of a surface of positive curvature
with A = 0 and C 00 = oo is the paraboloid

(M^2 ,g) ~ {(x,y,z) E JR^3 : z = x^2 +y^2 }.
OUTLINE OF PROOF. Short-time existence follows from the Bernstein-
Bando-Shi estimates. As with the Ricci fl.ow on compact surfaces (see Sec-
tion 3 of Chapter 5 in Volume One), long-time existence is proved by using a

potential f such that b..f = R (where b.. denotes the Laplacian with respect

to g) and examining the evolution of the quantity

h ~ R+ J'VfJ^2.

In fact, for metrics of the form (B.1) we can use f = -u. Because

au

• =b..u=-R
  at
  and

:t J'VuJ2 = b..J'VuJ2 - 2J\72uJ2'

we have

ah = b..h - 2JMJ2

at '

where M is the symmetric tensor with components

Mij = \7i\7ju + !Rgij·

Long-time bounds for J'VuJ and R (and higher derivatives) follow.

The bounds on JRJ imply that the metric remains complete; in particular,

the length of a given curve at time t > 0 is bounded above and below by

multiples of its length at time zero. By a theorem of Huber [210], the

hypothesis JR_ dμ < oo implies that JR dμ ::::; 47rX (M) on a complete

surface M. In particular, J JRI dμ is finite, and this is preserved by the

Ricci fl.ow. Finite total curvature, together with bounds on J\7 RI at any
positive time, imply that R decays to zero at infinity. One then shows that
C 00 (g) and A (g) are preserved under the fl.ow.

In the special case when R(go) > 0, limt->oo eu(x,y,t) exists pointwise and

is either identically zero or positive everywhere. In the latter case, au/ at =

• R implies that J~ Rdt is bounded, and hence the limiting metric is fiat. In
  the general case, we may define diffeomorphisms </>t(x,y) = e-u(O,O,t)f2(x,y),
  so that g(t) = ¢'tg(t) is constant at the origin. Then uniform bounds on