38 1. RICCI SOLITONS
The Ricci tensor of this metric is simply
Rc(g) = -8(w^1 0 w^1 ).
Given anyμ E JR, define a vector field
X = μ [-F1 - e-x^1 x3F2 + e-x^1 x3F3] + (1-μ) [F1 - ex^1 x2F2 - ex^1 x2F3].The computation
implies
-2Rc(g) = £xg + 4g,
and hence (S, g, X) is a Ricci soliton structure.
REMARK 1.58. As in the previous example, X -=/=grad f for any soliton
potential function f.
REMARK 1.59. Compact locally homogeneous manifolds with sol^3 ge-
ometry are mapping tori of YA : T^2 --+ T^2 induced by A E SL(2, Z) with
eigenvalues ,_ < 1 < .A+. As in the previous example, the soliton structure
cannot descend to any compact quotient, because the scalar curvature of g
is R= -2.
6.3. Type III singularity models. An important reason for studying
shrinking or steady solitons is that they can provide valuable information
about finite time singularities of Ricci flow. For example, the (Type 1)
neckpinch singularity is modeled in all dimensions n 2: 3 by the shrinking
gradient cylinder soliton
(JR x sn-l, g = ds^2 + 2 (n - 1) 9can, x = grad(s^2 /4)).
(See Section 5 in Chapter 2 of Volume One and [7, 8].) The conjectured
(Type II) degenerate neckpinch (Section 6 in Chapter 2 of Volume One) is
expected to be modeled on the Bryant soliton, discussed above. We shall now
see that homogeneous expanding solitons can model infinite time behavior
of Ricci flow.
DEFINITION 1.60. A Type Ill solution of Ricci flow (Mn,g(t)) exists for
t E [O, oo) (i.e., is immortal) and satisfies
sup ti Rm I < oo.
Mx[O,oo)There are many examples of Type Ill solutions (e.g., manifolds with
ni1^3 or sol^3 geometry) that collapse with bounded curvature. As t --+
oo, these examples exhibit pointed Gromov-Hausdorff convergence to
lower-dimensional manifolds. (See Sections 6 and 7 in Chapter 1 of Vol-
ume One.) For such solutions, it is not possible to form a limit solution
(M~,g 00 (t)) in a naive way. However, Lott has shown that such solutions
may have limits, properly understood, which turn out to be expanding ho-
mogeneous solitons [256]. We will describe only two of his results, omitting