1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

84 19. GEOMETRIC PROPERTIES OF ii:-SOLUTIONS

Now · -w(ro)^1 - 2 < - C^2 r- 0 1 and

llaf w' (ro + w (ro) r) drl ::::; C llaf (ro + w (ro) r)-^1 /^2 drl

= -( 2c -) I (ro + w (ro) r) l/2lfl

.. w ro. o
2Clfl
-.. ( + ( ) ro w ro r -)1/2 + r1/2.
0
Hence

(19.6)

l

w(ro+w(ro)r)_ 1 1< 2Clrl

w (ro) - (ro + w (ro) f)^1 /^2 + r6^12

for any ro E (O,oo) and r E (-w (ro)-^1 ro,oo ). Since w (ro)::::; Cr6^12 , this

shows that (see the next exercise) for any c: > 0, there exists r (c:) < oo such

that any point x E M with d (x, 0) > r (c:) is the center of a ball which is

c:-close in c^0 to a round unit cylinder of length c^1. (That is, we choose
C^1 ::::; r ~ d (x, 0)^112 .)

EXERCISE 19.13 (Estimates for curvatures of then-dimensional Bryant
soliton). Show the following.

(1) The estimates for w and w' in (19.5).

(2) Estimates for the sectional curvatures: There exist constants C E

(1, 00) depending only on the dimension n such that

c-^1 d (x, 0)-^1 ::::; V1 ::::; Cd (x, 0)-^1 '

c-^1 d (x, 0)-^2 ::::; V2::::; Cd (x, 0)-^2 '

where v1 and v2 denote the sectional curvatures of the planes tan-

gent to the spheres and the planes with one radial and one spherical

direction, respectively.

(3) Show that

c-^1 d (x, 0)-^1 ::::; III (x)I ::::; Cd (x, 0)-^1 '

where II (x) denotes the second fundamental form of the geodesic

(n - 1)-sphere centered at 0 passing through x.

HINT: See §4 of Chapter 1 in Part I. Also note that the second funda-

mental form of the level hypersurface 1-^1 (a) is given by II= T\7~{.

To show that the c^0 c:-neck in (Mn,Bry) obtained above is actually a

ck c:'-neck (where c:' tends to zero and k tends to infinity as c: tends to zero),

we need to estimate the higher derivatives; we leave this as an exercise.

EXERCISE 19.14. For the Bryant soliton (Mn,9Bry), show that for any
c: > O, there exists r (c:) < oo such that any point x EM with d (x, 0) > r (c:)
is the center of a ball which is an embedded c:-neck.