CHAPTER 33
N oncompact Hyperbolic Limits
And when your run is over jus t admit when it's at its end.- From "'Till I Collapse" by Eminem featuring Nate Dagg
In this chapter we discuss what happens when noncompact hyperbolic limits
occur in Case III for 3-dimensional nonsingular solutions to the normalized Ricci
fl.ow (NRF) (see Proposition 32.12 and the comments after its proof). The outline
of this chapter is as follows.
In §1 we state the main results that shall be proved in this chapter.In §2 we show that a noncompact hyperbolic limit (1i^3 ,h), after truncating
the cusp ends, corresponds to a time-dependent almost hyperbolic piece of the non-singular solution (M^3 , g ( t)) via h armonic diffeomorphisms which are approximate
isometries.
In §3 we prove the stability of any hyperbolic limit 1i. This result says that 1i
has a corresponding (time-dependent) stable asymptotically hyperbolic submanifold
of M which exists for all time and limits to 1i as t ---+ oo. This proof uses foliations
of the ends of almost hyperbolic pieces by constant mean curvature (CMC) tori.
In §4 we prove the crucial result that almost hyperbolic pieces a re necessarily
incompressible in the original manifold M.
We remark that there is an alternate proof of the stability of hyperbolic limits,
which is presented in §90 of Kleiner and Lott [161]. This proof avoids the use of
CMC surfaces and makes stronger use of the Mostow rigidity theorem.
There are also alternate proofs of the incompressibility of the cuspidal tori. In
§8 of Perelman [313], for solutions to the unnormalized Ricci fl.ow, his proof uses
the scale-invariant quantity A (t) Vol (g (t))^213 , where A (t) is the lowest eigenvalue
of the elliptic operator -4~g(t) + Rg(t)• In §93.1 of Kleiner and Lott [161], a
simpler argument using Rmin (t) Vol (g (t))^2 /^3 is given. Both of these proofs use the
existence of Ricci fl.ow with surgery.
In this chapter we shall say that a 3-dimensional nonsingular solution (M^3 ,
g(t)), t E [O, oo), to the NRF satisfies Condition H if it is in Case III with
the existence of at least one complete noncompact hyperbolic limit and if M is
connected, closed, and orientable.Since M is closed, any hyperbolic limit has finite-volume. Throughout this
chapter, each finite-volume hyperbolic limit (1i^3 , h) will be assumed to be
connected, orientable, complete, and noncompact. We also adopt the notations of
the previous chapter.
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