- GENERAL SOLITONS AND THEIR CANONICAL FORMS 3
By choosing O" (t) so that 0-(t) changes sign, this soliton may both expand
and shrink at different times.
On the other hand, in general, a Ricci soliton solution is evolving purely
by scaling (modulo diffeomorphisms). Since the time-derivative of the metric
is equal to the negative of twice the Ricci tensor, which is scale-invariant, it
is thus natural to ask whether one can put the general soliton equation in a
canonical form where O" (t) is equal to a linear function. The following gives
us a condition under which we may assume that a Ricci soliton defined as
in (1.1) has such a form.
PROPOSITION 1.3 (Canonical form for a general soliton). Let (Mn, g(t))
be a Ricci soliton, and assume that the solution of the Ricci flow with ini-
tial metric go = g(O) is unique among soliton solutions. Then there exist
diffeomorphisms 'lj;(t) : M--* M and a constant c: E ffi. such that
(1.5) g(t) = (1 +ct) 'lj;(t)*go.PROOF. Differentiating (1.3) with respect to time gives(1.6) 0-( t)go + .C X(t)go = 0.
Case 1. If 0-(t) = O, then O"(t) = 1 + c:t for some constant c:. Hence, by
(1.1), we may simply take 'lj;(t) = 1.p(t) to obtain (1.5).Case 2. If 0-(t) is not identically zero, then let Yo= -X (to) /0-(to) at some
to where 0-(to) #-0. We then have
(1. 7) .Cyogo = go.
Substituting (1.7) into (1.3), we have
-2Rc(go) = .C&(t)Yo+X(t)go
for all t. Consider the vector field
Xo ~ 0-(0)Yo + X(O).
Then
-2 Rc(go) = .Cxago.
Let 'lj;(t) be the 1-parameter group of diffeomorphisms generated by Xo.
Then it is easy to check that g(t) = 'lj;(t)*go satisfies the Ricci flow with the
same initial conditions go and is a steady soliton. Thus, by our uniqueness
assumption for soliton solutions to the Ricci flow with initial metric go, by
replacing 1.p(t) by 'lj;(t), we have O"(t) = 1 in (1.1). D
REMARK 1.4. The proof shows that under the uniqueness assumption,
a Ricci soliton not in canonical form can be made a steady soliton.
If (Mn,g(t)) is a Ricci soliton where (1.5) holds, then we say that the
soliton is in canonical form. By rescaling, we may assume that c: = -1, 0,
or 1; these cases correspond to solitons of shrinking, steady, or expanding
type, respectively.