1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

32 1. RICCI SOLITONS

for some constant C > 0. Because dr = Wds and W = 0(1/r), then

d w^2

-log-< cr-^3

dr Y - '

from which it follows that W^2 /Y is bounded above for large r. Thus, there

are positive constants c1, c2 such that

w2

- <X <c1W^2 ,

ci

w2.

• <Y<c2W^2
  c2

for sufficiently larger, and the bounds on X/Y follow. D

Much more detailed information about the solutions in the positive and

negative curvature cases (as well as the steady case) has been obtained

by Robert Bryant (in unpublished notes, using different coordinates). In

particular, Bryant proved

PROPOSITION 1.43 (Bryant). Each positive curvature solution defines a

complete, rotationally symmetric expanding soliton, whose sectional curva-

ture decays proportionally to 1/r^2.

6. Homogeneous expanding solitons

6.1. Existence. Homogeneous spaces are among the nicest examples

of Riemannian manifolds.^13 Einstein metrics are among the nicest examples

of Riemannian metrics. It is thus natural to ask if a given homogeneous

space Mn = G / K admits a G-invariant Einstein metric. If M is closed,

the answer is frequently 'yes'. For example, consider the following result of

Bohm and Kerr [29].

THEOREM 1.44. Let Mn be a closed, simply-connected homogeneous

space. If n < 12, then M admits a homogeneous Einstein metric.

On the other hand, Wang and Ziller's example SU(4)/ SU(2) of a 12-

dimensional homogeneous space that admits no homogenous Einstein metric

shows that the answer is 'no' in general and that the dimension restriction

above is sharp [365].

Every known example of a noncompact, nonfl.at homogeneous space that

admits an Einstein metric is isomorphic to a solvable Lie group S. (See

[198] and [317].) Moreover, for all known examples, the Einstein metric is

of standard type.

DEFINITION 1.45. A left-invariant metric g on a solvable Lie group S,

regarded as an inner product on the Lie algebra .s, is said to be of standard

type if the orthogonal complement with respect to g of the derived algebra

[.s,.s] forms an abelian subalgebra a of .s.

(^13) See Chapter 7 of [27], for example.