246 8. DILATIONS OF SINGULARITIES
Thus we may replace IRml by R in the discussion above, whence it follows
that
Roa (x, t) :S 1 =Roa (xoa, 0)
for a pointed Type II limit (M~,goa (t) ,xoa)- The advantage of the scalar
curvature Roa attaining its maximum in space and time is as follows. A
stronger consequence of Theorem 9.4 is Corollary 9.7, which says that any
type of limit of dilations about a finite-time singularity has a non-negative
curvature operator. When R 00 attains its maximum, this enables the appli-
cation of a differential Harnack estimate together with the strong maximum
principle.
EXAMPLE 8.23. An example of a rotationally symmetric Type II limit
is the Bryant soliton [21] on JR^3.
- Dilations of infinite-time singularities
5.1. Type II limits of Type Ilb singularities. This case is dual to
the Type Ila case. Recall that the Type Ilb condition is
(8.21) sup t IRm (x , t)I = oo.
Mnx[O,oo)Analogous to Type Ila, choose any sequence Tj / oo. If we follow what we
did before, it is natural to choose a sequence (xi, ti) such that
ti IRm (xi, ti)I ~ C:Xi---+ ooand
ti IRm (xi, ti)I
---------> c > 0.
supMnx[O,Ti] t IRm (x, t)I -
This is always possible but may not be a good choice, because it does not
guarantee that the curvatures of the dilated solutions can be uniformly
bounded on finite time intervals. To see this, notice that the dilated so-
lution 9i ( t) is defined for t E [-ai, oo) and satisfies
IRmi(x, t)I = IRm (~i, ti) I [Rm ( x, ti+ IRm (~i, ti)I) I
=1
ti IRm (xi, ti)I I Rm (x, ti + IRm (xi, t ti)I ) I
(
x ti+---t - ) ti IRm (xi, ti)I
IRm (xi, ti)I ti IRm (xi, ti)I + t
1 C:Xi
(8.22) < ---- c C:Xi + t
for all x E Mn and t such that
0 :S ti+ t IRm (xi, ti)l-^1 :S Ti,
hence for all t E [-ai, wi], where
Wi ~(Ti - ti) IRm (xi, ti)I.