1547671870-The_Ricci_Flow__Chow

14. MONOTONICITY OF THE ISOPERIMETRIC CONSTANT 157

REMARK 5. 78. The result that a sequence of blow-ups (M^2 , 9J ( t)) pulled

back by diff eomorphisms converges smoothly to a shrinking round sphere
metric (5^2 ,9 00 (t)) is weaker than the pointwise C^00 convergence obtained
above. Moreover, if one did not already know the Uniformization Theorem,
one could not conclude that 900 is in the same conformal class as the origi-
nal solution. An analogous problem occurs when studying the Kahler- Ricci
fl.ow in complex dimensions greater than one: one does not know in gen-
eral whether or not a limit 900 is Kahler with respect to the same complex
structure as the original solution.

The strategy of proving sequential convergence 9J (t) ____, 900 (t) uses sev-
eral techniques that will be developed later in this volume. We provide the
details in Section 15. In this section, we develop the background necessary
to obtain a suitable injectivity radius estimate.

DEFINITION 5. 79. Let (Mn, 9) be a closed orientable Riemannian man-

ifold. We say a smooth embedded closed (but possibly disconnected) hy-

persurface "En-l C Mn separates Mn if Mn\"En-l has two connected

components M1 and M2 such that

8M1 = 8M2 = "En-^1.

DEFINITION 5.80. If "En-l separates Mn, the isoperimetric ratio of

"En- l in (Mn, 9) is

n-l. n-l n 1

(

1

)

n-l

Cs ("E ) 7 (Area ("E ) ) Vol (M1) + Vol (M2)

The isoperimetric constant of (Mn, 9) is

where the infimum is taken over all smooth embedded "En- l that separate

Mn.

REMARK 5.81. Cs ("En-l) is invariant under homothetic rescaling of the

metric 9 1--t e>..9 and is equivalent to the standard isoperimetric ratio

C "En-1 == (Area ("En-1) r.

I( ) · min {Vol (M1), Vol (M2)}n-l

Indeed, one has

Now we restrict ourselves to the case of a closed orientable Riemannian

surface (M^2 , 9). If/ separates M^2 into two open surfaces Mi and M§, we