- THE MAIN RESULT ON NONSINGULAR SOLUTIONS 201
(12) A rotationally symmetric n eckpinch (see Angenent and one of the authors
[13]).
(13) A rotationally symmetric degenerate neckpinch (see Gu and Zhu [127]).2.2. Statement of the main result on 3-dimensional nonsingular so-
lutions.
The following is Hamilton's classification of 3-dimensional nonsingular solutions
[143].THEOREM 32.2 (Nonsingular solutions on 3-manifolds admit geometric decom-
positions). If a closed 3-manifold M^3 admits a nonsingular so lution of the NRF,
then M is diffeomorphic to one of the following:
(Al) a graph manifold,^4
(A2) a spherical space form S^3 /r,
(A3) a fiat manifold,
(A4) a hyperbolic manifold,
(A5) the union along incompressible 2-tori of finite-volume hyperbolic manifolds
and graph manifolds.In particular, in all cases M admits a geometric decomposition in the sense of
Thurston.^5
We say that a nonsingular solution of the NRF sequentially collapses if there
exists a sequence of times ti~ oo such that the metrics g(ti) collapse; i.e.,(32.2) lim p (ti)
2max IRml = 0,
i_,oo Mx{t; }
where
p(ti) ~ ~~inj 9 (t,) (x)is the maximum injectivity radius of any point for the metric g(ti) (compare with
(31.15)). Otherwise, we say that the solution g(t) is noncollapsed.
As we shall exposit, Hamilton proved that, under the hypothesis of Theorem
32.2, exactly one of the following cases occurs, corresponding to the five cases in
the statement of the above t heorem:
(Bl) (Sequentially collapses) It then follows from Cheeger- Gromov theory
that M admits an F-structure and is diffeomorphic to a graph manifold (see
Theorem 31.55).
In the rest of the chapter we sh all assume that g(t) is noncollapsed.Thus there exists a constant 6 > 0 such that for each time t E [O, oo) there exists
a point Xt E M such that inj 9 (t) (xt) ;:::: 6. This enables us to apply Hamilton's
Cheeger- Gromov-type compactness theorem to obtain a limit solution g 00 (t) (see
Theorem 32.5 below).
(B2) (Exponential convergence to a spherical space form) The solution
g(t) converges exponentially fast in every Ck-norm a8 t ~ oo to a static constant
positive sectional curvature metric g 00 on M.
(^4) For the definition and topological classification of graph manifolds, see §5 in Chapter 3 1.
(^5) For the definition of geometric decomposition, see Thurston [402]. This is a lso briefly
discussed in C hapter 1 of Volume One and Chapter 9 of Part I.