120 19. GEOMETRIC PROPERTIES OF 11;-SOLUTIONSSince the RHS tends to zero as 11; 1 tends to zero, this inequality gives a
uniform positive lower bound for 11; 1 and thus the theorem is proved. D
In summary, a 3-dimensional noncompact 11;-solution has an asymptotic
shrinking soliton which is a round cylinder 52 x JR: or its (noncompact)quotient (5^2 x JR:) j'll 2. This corresponds to c-necks in the 11;-solution at
times ti= -Ti --+ -oo, where the reduced distance and scalar curvature are
controlled. This leads to a positive lower bound for the reduced volume at
Ti independent of i.7. Notes and commentary
§1. 11;-solutions arise as singularity models in the finite time singularity
analysis of the Ricci flow on closed 3-manifolds. We remark that in dimen-
sions greater than or equal to 4, one may wish to study ancient solutions
which are 11;-noncollapsed on all scales and have curvature conditions which
are no weaker than nonnegative scalar curvature and no stronger than non-
negative curvature operator. For, on one hand, ancient solutions must have
nonnegative scalar curvature; note also that any Einstein solution with pos-
itive scalar curvature is 11;-noncollapsed at all scales. On the other hand,
Bohm and Wilking have shown that the singularity models of solutions of
the Ricci flow on closed manifolds whose initial metrics have 2-nonnegative
curvature operator are shrinking spherical space forms.
§2. Regarding AVR and ASCR under Ricci flow, we refer the reader to
§§18, 19, and 22 of Hamilton's [92] and Chapter 8 of [45].
§3. In dynamical systems, a heteroclinic orbit is a path which joins
two different stationary (equilibrium) points. In the space of metrics modulo
diffeomorphisms and scalings on a closed manifold, a metric is a stationary
point of the Ricci flow equation if and only if it is a Ricci soliton.
QUESTION 19.59. Are there examples of ancient solutions which are not
heteroclinic orbits, where convergence of metrics is defined in the Cheeger-
Gromov sense and where collapsing is allowed (in particular, the topology
and dimension of the underlying manifold is allowed to change in the limit)?
For example, one may think of the King-Rosenau ancient solution ( a.k.a.
sausage model) as joining a pair of cigar solitons, with a flat cylinder in
between, at time -oo to a round sphere at time 0 (the solution shrinks to
a round point). Similarly, Perelman's ancient solution on then-sphere joinstwo Bryant solitons, with a round cylinder in between, at time -oo to a
round point at time 0. Note that physicists have defined other examples of
ancient solutions.
REMARK 19.60. One may also ask the above question assuming theancient solutions are 11;-noncollapsed at all scales for some 11; > 0.
§5. Here we give a first variation calculation related to the £-length.
Let g ( r) be a solution to the backward Ricci flow ff 7 g = 2 Re. We consider