1. GENERAL SOLITONS AND THEIR CANONICAL FORMS 3 By choosing a (t) so that &(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 a (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(0) is unique among soliton solutions. Then there exist diffeomorphisms ip(t) : M M. and a constant £ G R such that (1.5) g(t) = (l + et)iP(t)*9o- PROOF. Differentiating (1.3) with respect to time gives (1-6) &(t)g0 + C~{t)90 = 0. Case 1. If a(t) = 0, then a(t) = 1 + et for some constant e. Hence, by (1.1), we may simply take i^(t) = p(t) to obtain (1.5). Case 2. If a(t) is not identically zero, then let 1Q = —X (to) /cr(*o) at some to where a (to) ^ 0. We then have (1.7) CYogo = #o- Substituting (1.7) into (1.3), we have -2Rc(g0) = £&{t)Yo+x{t)9o for all t. Consider the vector field X0 = &(0)Yo + X(0). Then -2Rc(g0) = CXogo. Let ip(t) be the 1-parameter group of diffeomorphisms generated by XQ. Then it is easy to check that g(t) = ip(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 /?(£) by ip(t), we have a(t) = 1 in (1.1). 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 £ = —1,0, or 1 these cases correspond to solitons of shrinking, steady, or expanding type, respectively.
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