238 8. DILATIONS OF SINGULARITIES
In order to develop intuition for the dilation criteria introduced below,recall that if the singularity time Tis finite, then Theorem 6.54 tells us that
(8.3) lim suplRm(-,ti)I = ooti/T Mn
regardless of whether the singularity is forming quickly (Type I) or slowly
(Type Ila). In light of this fact, one may choose the sequence of times
ti so that the maximum of the curvature at time ti is comparable to its
maximum over sufficiently large previous time intervals. Similarly, one may
choose the sequence of points Xi so that the curvature at (xi, ti) is com-
parable to its global maximum at time ti. Occasionally (for example, in
dimension reduction) one chooses points Xi so that the curvature at (xi, ti)
is comparable to its local maximum over a sufficiently large ball. (In some
cases, it may be possible to get a complete limit for any sequence (xi, ti)
such that I Rm (xi, ti)I ---+ oo if one has good enough bounds on IV Rml; we
shall not discuss such limits in this volume, however.)
Motivated by these considerations, we adopt the following definitions
for any solution (Mn,g (t)) of the Ricci fl.ow on a maximal time interval
0 ::; t < T ::; oo. Notice that they reflect the way parabolic PDE naturally
equate time with distance squared.
DEFINITION 8. 10. We say a sequence (xi, ti) is globally curvature
essential if ti / T ::; oo and there exists a constant C 2: 1 and a sequence
of radii Ti E ( 0, Vii) such that
(8.4) sup {!Rm (x, t)I: x E Bg(t) (xi, Ti), t E [ti - TT, ti]} ::; C IRm (xi, ti)Ifor all i E N, where
(8.5) i-too .lim TT !Rm (xi, ti)I = oo.
DEFINITION 8.11. We say a sequence (xi, ti) is locally curvature es-
sential if ti / T ::; oo and there exists a constant C < oo and a sequence
of radii Ti E (0, Vii) such that for all i E N, one has
(8.6) sup { IRm (x, t) I : x E Bg(t) (xi, Ti), t E [ti - TT, ti] } ::; CTj^2.
In this volume, we shall deal mainly with globally essential dilation se-
quences. In a later volume, we hope to discuss the relevance of locally
essential sequences to the surgery arguments in [ 106 ].
REMARK 8.12. In some cases, one works with a special case of condition
(8.4) in which
(8.7) sup { IRm (x, t) I : x E Mn, t E [O, ti]} :SC IRm (xi, ti)I.
REMARK 8.13. By (8.4), any smooth limit of a globally curvature es-
sential sequence formed by the parabolic dilations (8.8) has its curvature
bounded by some function K (t). Moreover, property (8.5) ensures that any
such limit is complete. A smooth limit of a locally curvature essential se-
quence will have bounded curvature on a metric ball of radius 1 for times