1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

is complet e and satisfies 0 ::::; R 9 00 (t) ::::; C for t E ( -oo, oo). By Theorem 28.41
together wit h t he work of one of the aut hors in [83], since we h ave a n eternal

solut ion with nonnegative bounded curvature, we conclude that g 00 (t) is either a

cigar soliton or the flat plane. Proposit ion 29 .38 now follows from Lemma 29. 3 1.

11. Irreducible components of \7^3 v

Let ( 52 , g ( t)), t E ( -oo, 0) , be an an cient solution of the Ricci flow on a m axi-
mal t ime interval. The trace-free symmetric p art of the third cova riant derivative of
the pressure function v h as a remarkable property. Na mely, the quantity Q , equal-
ing v times t he norm squared of t his tensor, is a subsolution to the heat equation.
We shall prove this by long but straight forward computations in § 12 of this chapter.
Moreover , in §13 we show that a nonround g(t) is the King- Rosen au solution if and
only if Q vanishes. The main theorem of this ch apter then follows from proving
tha t Q = 0. This will b e done toward t he end of t he ch apter.

11.1. The quantity Q.

Let { x^1 , x^2 } b e local coordinat es on 52. For a covari ant 3-tensor

2

a = L aiJkdxi ® dxi ® dxk

i,j,k=l

which is symmetric in j and k , its symmetrization is give n by

1 1

(29.127) S(a )iJk ~ 3a[ijkJ = 3 (aijk + ajki + akiJ),

where [ij k] here and b elow represents cyclically p ermuting i, j , and k and summing.

Let v(t) b e the pressure function of g(t). The third covariant derivative \7~Jkv =

\7^3 v ( 0 ~;, 8 ~ 1 , D~k) is symmetric in its last two compon ents, where \7 denotes the
Riemannia n covaria nt deriva tive of g 5 2. Define the symmetric 3-tensor

(29.128)

By the commutator formulas for covariant derivatives , we h ave (lowering indices
and summing over repeat ed indices)

(29.129a)

(29.129b)

3 1 5 2 5 2

BiJk = \7iJkv - 3(RJike\7ev + RkiJe'Vev).

B iJk = \7~Jkv + A iJk,

1

Aijk ~ -3 (\7Jv(g 52 )ik + \7kv(g52)iJ - 2\7iv(g 52 )Jk).