1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

40 1. RICCI SOLITONS

With respect to the frame a(t), one obtains the identification

(

~r4/3 o o )

9a(t) = 0 1r2/3 0.

0 0 lt-2/3

The limit soliton metric is

9oo(t) = 3tga(t)·

EXAMPLE 1.64. Let (S, g, X) denote the sol^3 Ricci soliton structure of
Example 1.57. With respect to the frame (3 = (F1, F2, F3) constructed there,
regard the fixed metric g as the matrix

Yf=(H ~)

Now consider the time-dependent frame a(t) = (3A(t) given by

(

0 _l 0 )

A(t) = a(t) 0

2

O ,

0 0 a(t)

a(t) ={ft.

With respect to the frame a(t), one gets the identification

9cx(t) = 4 0 1 0.

(

1r1 o o )

o o lr 4 1

The limit soliton metric is

9oo(t) = 4tga(t)·

Lott also discovered a 4-dimensional example [256]. We will describe

its (time-independent) Ricci soliton structure, leaving construction of the

corresponding Ricci fl.ow solution as an interesting exercise for the reader.

EXAMPLE 1.65. Let N denote the simply-connected 4-dimensional nilpo-
tent Lie group nil^4 with bracket relations

and all other [Fi, Fj] = 0.

The frame field defined in standard coordinates (x1, x2, x3, x4) on JR^4 by

a a a a a a

F1=-

8

, F2=-

8

, F3=-

8

, F4=x1-

8

+x2-

8

+-

X1 X2 X3 X2 X3^8 X4

realizes these relations. The connection 1-forms are

(\lgF-) = ~ (-~4 -t

4

-~4 F1 ;_ F3).

• J 2 0 -F4 0 F2
  
  
  • F2 F1 - F3 F2 0