1547671870-The_Ricci_Flow__Chow

158 5. THE RICCI FLOW ON SURFACES

have

Cs (r) =Lg (r) 2 ( Ag (Mi)^1 +Ag (M§)^1 )

(5.47)

2 Ag (M2)

=Lg (r) Ag (Mi) ·Ag (M§)'

where Lg denotes length and Ag denotes area measured with respect to g.

Moreover, the infimum in Definition 5.80 is attained by a smooth embedded

curve.

LEMMA 5.82. If (M^2 , g) is a closed orientable Riemannian surface, there

exists a smooth embedded (possibly disconnected} closed curve/ such that

Cs (r) = Cs (M^2 ,g).

For a proof, see Theorem 3.4 of [ 69 ]. Analogous results in general di-

mensions can be found in Theorems 5-5 and 8-6 of [99].

REMARK 5.83. The minimizing curve r might be disconnected, as can

easily be seen by considering a rotationally symmetric 2-dimensional torus

with two very thin necks. In this case, / is the union of two disjoint embed-

ded loops. More generally, no connected embedded homotopically nontrivial

curve will separate a torus.

Now we restrict ourselves further to the case that ( M^2 , g) is diffeomor-

phic to 52. Define a loop in M^2 to be a connected closed curve. It follows

from the Schoenfiies theorem that any smooth embedded loop r separates

M^2 into smooth discs Mi and M§. We denote the isoperimetric ratio of a

smooth loop on a topological sphere by

Then we define

CH (M^2 ,g) ~inf CH (r),

'Y

where the infimum is taken over all smooth embedded loops r· Clearly,

Cs (M^2 ,g):::; CH (M^2 ,g).

If ( 52 , 9can) is a constant-curvature 2-sphere, then CH ( 52 , 9can) = 47r.

Moreover, this is an upper bound.

LEMMA 5.84. If (M^2 , g) is a closed orientable Riemannian surface, then

CH (M^2 , g) :::; 47r.

PROOF. Let x E M^2 be given and define

le~ {y E M^2 : distg (x,y) = c}.