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254 33. NONCOMPACT HYPERBOLIC LIMITS

It follows from Lemma 33.37 that

L (0) s (1 + c)^2 (1+2c) (1 + e-P• L (P•)) Lo

S ( 1 + c)^2 ( 1 + 2c) ( 1 + ( 1 + 2c) e -P • L ( 0)) Lo.

By choosing A small , we can make Lo as small as we like. This implies that L (0)

is as small as we like:

PROPOSITION 33.40 (Length is small). For any E1 > 0, there exists Ao E (0, A)

such that for any A S Ao we have the following. Let c: be chosen sufficiently small

and let tc and T 2 c be as in Lemma 33.37. Choose t~ < oo so that the inequalities

(33.82), (33.83), and (33.84) hold for t :?: t~. If t<> :?: max{ tc, T2c ,tD is such that

(33.62) holds, then any area minimizing disk V (t<>) in iA (t<>) satisfies

(33.86) L g(to) ( 8V ( t<>)) S c1.

We are now ready to prove the following.

LEMMA 33.41. Let E > 0 be sufficiently small. If A is sufficiently small and

t<>:?: max{tc, T2c, tD is such that (33.62) holds, then

d+ A

dt JA (t<>) s -27f + €.

PROOF. Applying Proposition 33.40 with c 1 = c/3 to (33.58) yields

d+ AJ

~(t<>) S -27r + (1+2c) L g(to) (8V(t<>)) - (1 - c) A J(t<>)

s -27f + €.

By this and Lemma 33.30, we obtain Proposition 33 .24.

0

At this point, to complete the proof of Hamilton's theorem, Theorem 32.2, it
only remains to prove Propositions 33.11 and 33.12. We shall accomplish this in
the next chapter using the inverse function theorem.

5. Notes and commentary

§1. Definition 33.l is Definition 9.2 in [143]. Proposition 33.12 is Theorem 9.1
in Hamilton [143].

§2. Proposition 33.13 is Theorem 9.4 from [143]. Proposition 33. 14 is Theorem
9.3 in [143].

§3. We followed §10 of [143] closely in proving the stability of hyperbolic limits
of 3-dimensional nonsingular solutions of the Ricci fl.ow.
Another, albeit one more complicated to justify, way to obtain (33.26) is as
follows. Observe that for i sufficiently large, we have that every totally umbillic

CMC torus of area A/2 in il is contained in (h Since i : CUi, h) ---+ (Yi, g (.Bi) Iv,)
is an almost isometry, we have that (Yi, g (.Bi)lv,) is an almost hyperbolic piece in

Fi (,Bi ) (HA) C '\Ii.