1547671870-The_Ricci_Flow__Chow

236 8. DILATIONS OF SINGULARITIES

the curvature bound

sup IRmoo(·, t)l · ltl < oo.

M~x(-oo,O]

• One says ( M~, 900 ( t)) is an ancient Type II singularity model
  if it exists on a time interval ( -oo, w) containing t = 0 and satisfies
  the curvature bound

sup I Rm oo(-, t) I · ltl = oo.

M~x(-oo, 0 ]

• One says (M~, 900 (t)) is an eternal Type II singularity model
  if it exists for all t E (-oo, oo) and satisfies the curvature bound

sup IRmoo(-, t)I < oo.

M~ x (- 00,00)

• One says (M~, 900 (t)) is an immortal Type III singularity

model if it exists on a time interval (-a, oo) containing t = 0 and

satisfies the curvature bound

sup IRm 00 (-, t)I · t < oo.

M~x[O,oo)

REMARK 8.4. As we shall see when we study dilation below, it is always
possible to apply a more careful 'point picking argument' and thereby to
choose the sequence of points and times about which one dilates so that the
singularity model satisfies

(8.1) sup IRmoo (x,O)I = IRmoo (y,O)I

xEMn

for some y E Mn. If the curvature operator of the solution (M~, 900 (t)) is
nonnegative, one can replace this with

(8.2) sup R 00 (x, 0) = R 00 (y, 0).

xEMn

The latter condition allows application of the strong maximum principle
and a differential Harnack inequality. (We introduced Harnack estimates in
Section 10 of Chapter 5 and will see them again in Section 6 of Chapter

9. We will make a thorough study of differential Harnack estimates in the
   successor to this volume.) Condition (8.2) is useful primarily for Type II
   singularities. (For example, see Section 6.)

REMARK 8.5. By Theorem 9.4, any limit (M~, 900 (t)) of a finite-time
singularity in dimension n = 3 has nonnegative curvature operator. So in
dimension 3, one can always satisfy (8.2) by making appropriate choices of
points and times about which to dilate.