1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. 2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH 51

Choosing co E (0, oo) so that1-l^2 (f-^1 [0, c 0 )nB(r)) = 1-l^2 (f-^1 [c 0 , oo)nB(r)) = ~7fr^2 ,

we have that

REMARK 28.38. See General Isoperimetric Theorem 4.4.2 in [ 11 0] for a general

result that encompasses relative isoperimetric inequalities, but without the sharp

constants.

3.2. Width.

Let (M^2 , g) be an orientable complete noncom pact Riemannian surface. Given

a C^00 proper function F: M---+ [O,oo), its width w(F, g) with respect tog is

defined to be the supremum of the lengths of the level sets of F; i.e.,

(28.37) w (F, g) ~ sup L 9 (F-^1 (c)),

cE(O,oo)

where L 9 (S) = 1-l^1 (S) E [O, oo] denotes the 1-dimensional Hausdorff measure of a

subset SC M. Note that SC M is the image of a rectifiable curve if and only if

S is compact, connected, and has L 9 (S) < oo.

DEFINITION 28.39. The width of the metric g is the infimum of the widths

of all C^00 nonnegative proper functions F with respect tog; i.e.,

(28.38) w (g) =inf w (F, g) E [O, oo].

F
The Euclidean plane (JR^2 , dx^2 +dy^2 ) has infinite width. To see this, note that for
any smooth proper function F , the set p-^1 (c) bounds the compact set p-^1 ([0, c]).
By Sard's theorem, almost every c is a regular value of F, in which case p-^1 (c) is
a (possibly disconnected) smooth curve. By the Euclidean isoperimetric inequality,
we have

(28.39)

for such c; the RHS approaches infinity as such c ---+ oo. In fact, by t he more
general isoperimetric inequality for currents (see Theorem 4.5.9(31) in Federer's
book [ 11 0]), we have that (28.39) holds for all c E [O, oo).
We have the following property of width under Cheeger- Gromov convergence.

LEMMA 28.40 (The limit of surfaces with uniformly bounded widths cannot be

Euclidean). Let (MI, gi, Pi) be a sequence of complete Riemannian surfaces which

converge in the pointed C^00 Cheeger-Gromov sense to (M~, g 00 , p 00 ). If the widths

of (Mi, gi, Pi) are bounded independent of i, then (M 00 , g 00 ) cannot be isometric to

the Euclidean plane.

PROOF. Suppose that (M 00 , g 00 ) is isometric to the Euclidean plane. Then,

for any k EN, there exists i(k) EN such that Bk~ B 9 ,(k) (Pi(k)i k) C Mi(k) is k-^1 -

close in the Ck-norm to the Euclidean 2-ball of radius k. Let Fk: Mi(k) ---+ [O, oo)

be any C^00 proper function satisfying w (Fk, gi(k)) < oo. The function

Gk: [O,oo)---+ [O,Area 9 ,(k)(Bk)]

defined by