- HOMOGENEOUS EXPANDING SOLITONS 35
Lauret also gives a variational characterization of Ricci solitons [244].
Roughly speaking, this says that any simply-connected nilpotent Lie group
( N, 9) admitting a Ricci soliton structure is a critical point of a functional
that measures, in a certain sense, how far ( N, 9) is from being Einstein.
We now make this precise. It will be more convenient to fix an in-
ner product and allow the Lie algebra brackets to vary. (Compare with
Examples 1.63 and 1.64 below.) Specifically, fix an n-dimensional inner
product space ( n, ( ·, ·)) and let A = A^2 ( n*) ® n denote the space of all skew-
symmetric bilinear forms on n. (Compare to C2 2 im(b"i) defined above.)
Let N denote the subspace of all nilpotent elements of A that satisfy the
Jacobi identity. Then N may be regarded as the space of all nilpotent Lie
brackets on n. To eachμ EN, associate the simply-connected Lie group Nμ
and the left-invariant metric 9μ on Nμ determined by (-, ·). Notice that N
is invariant under the natural action of the group GL(n). Thus the GL(n)
orbit ofμ EN corresponds to all simply-connected homogeneous nilpotent
Lie groups N isomorphic to Nμ, and the O(n) orbit ofμ corresponds to all
(N,9) isometric to (Nμ,9μ)·
The fixed metric (·, ·) on n induces a metric (·, ·) on A defined by(μ, v) = L (μ(Xi, Xj ), Xk) (v(Xi, Xj ), Xk)
i,j,kwhen {Xi, ... , Xn} is any orthonormal basis of n. Let
Ni={μ EN:(μ,μ)= l}.
Using the Ricci endomorphism, Lauret defines a functional F: Ni--+~ by
F(μ) = tr(Rc(9μ)^2 )
and proves
THEOREM 1.53. Let μ E Ni. Then (Nμ, 9μ) satisfies (1.64)-(1.66)
(i.e. admits a Ricci soliton structure) if and only ifμ is a critical point
of F.To interpret this result, observe that E(μ) = [Rc(9μ) - ~R(9μ)I[
2
mea-
sures the trace-free part of the Ricci endomorphism, i.e., how far 9μ is from
being Einstein. But for μ E Ni, one has
1
E(μ) = tr(Rc(9μ)^2 ) - -R(9μ)^2
n
1
=F(μ)--.
16nHence a local minimum μ of F in Ni should correspond to a homogeneous
space (Nμ, 9μ) that is closest to Einstein among all nearby (Nv, 9v ).