1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. SINGULARITY MODELS AND 11:-SOLUTIONS 83

As a higher-dimensional analogue of the 2-dimensional cigar soliton so-

lution we have the following.

EXAMPLE 19.12 (Bryant solitons). The Bryant soliton (Mn, 9Bry), where

n 2: 3 and Mn ~ JR.n, is the unique (up to scaling) complete nonfiat rotation-

ally symmetric steady gradient Ricci soliton (see §4 of Chapter 1 in Part I).

The 3-dimensional Bryant soliton is useful to keep in mind while developing

intuition in the study of singularity formation in dimension 3. In particu-

lar, it occurs as a singularity model for degenerate neckpinch singularities^3

(for a proof of this, see Gu and Zhu [84]; for some heuristic discussion, see

also §6 of Chapter 2 in Volume One and §2.3 of Chapter 8 in [45]). Note

that the sectional curvatures of the Bryant soliton decay inverse linearly

and quadratically, depending on the directions of the 2-planes (see Exercise

19.13 below). On the other hand, the curvature of the cigar soliton decays

exponentially.

Consider the Bryant soliton at a fixed time, i.e., Mn ~ JR.n with the

rotationally symmetric metric

9Bry ~ dr^2 + W (r)^2 9sn-i.

Here w together with a radial potential function f satisfies the ODE ( ordi-

nary differential equation) corresponding to the steady gradient Ricci soliton

equation Rc(g) + \1\1 f = 0 (see (1.45) in Part I):

II

f

l/ --n-, w ww' f' = ww" - (n - 1)(1 - ( w') (^2) ).
w
We have (see p. 24 in Part I)
(19.5) 0 s w' (r) s cr-^112 and ]:_rc^112 < -w (r) < -Cr^112

for some constant CE (0, oo) and all r E (0, oo). When considering ancient

solutions, we may keep in mind the following property of the Bryant soliton.

Existence of c-necks in the Bryant soliton.^4 Given ro E (0, oo),

consider the rescaled metric

-2 -2 (w (ro + w (ro) r))

2

go = w (ro) 9Bry = dr + w (ro) 9sn-1,

where r ~ w (ro)-^1 (r - ro).^5 We have

lw(ro;(~o~ro)f) -11 S llaf w'(ro+w(ro)r)drl.

(^3) Note also that given a rotationally symmetric Type Ila singular solution on sn, there
is a corresponding rotationally symmetric steady gradient soliton singularity model, which
must be the Bryant soliton. Numerical evidence of this was first given by Garfinkle and
one of the authors [67], [68].
(^4) See Definition 18.26.
(^5) The effect of the translation by -r 0 given in the change of coordinates defining f is
the same as choosing the center of the neck to have distance ro from 0 with respect to
9Bry·