4 1. RICCI SOLITONS A way to circumvent the undesirableness of the uniqueness assumption in Proposition 1.3 is as follows. Since the geometry of any Ricci soliton g(t) is the same as that of go, we will start with go and then construct another Ricci soliton in canonical form and with the same initial metric go. Choose any time to and let e = &(to) and Xo 4 = X(to)1 so that equation (1.3) becomes at t = to - 2 Rc(go) = ego + £x09o- We will now drop the subscripts on X and g. Using indices, the equation above now reads (1.8) -2Rij = ViXj + VjXi + egij, where X{ gijXi are the components of Xb, the covariant tensor (1-form) obtained from X by lowering indices using g. A triple (g, X, e) (or pair (g,X) if we suppress the dependence on e) consisting of a metric and a vector field that satisfies (1.8) for some constant e is called a Ricci soliton structure.2 We say that X is the vector field the soliton is flowing along. REMARK 1.5 (Solitons and normalized Ricci flow). If (g,X) is a Ricci soliton structure on a compact manifold M, then g evolves purely by dif- feomorphisms under the normalized (constant volume) Ricci flow. DEFINITION 1.6 (Gradient Ricci soliton). A Ricci soliton structure (#, X) is a gradient soliton structure if there exists a function / (called the po- tential function) such that Xb = df. In this case, (1.8) becomes (1.9) Rij + ViVjf + ^9ij = 0. The following, whose proof is elementary, shows that given a complete gradient Ricci soliton structure, we can construct a gradient Ricci soliton in canonical form (see Theorem 4.1 on p. 154 of [111] or Kleiner and Lott [231]). In particular, the result below illustrates the sense in which a Ricci soliton structure may be regarded as initial data for a Ricci solution, i.e., for a self-similar solution to Ricci flow. PROPOSITION 1.7 (Gradient soliton structures and canonical forms). Suppose (go, V/o,£) is a complete gradient Ricci soliton structure on Mn. Then there exists a solution g (t) of the Ricci flow with g (0) = go, diffeomor- phisms (p(t) with p(0) = id^, and functions f (t) with f (0) = fo defined for all t with (1.10) r(£) = l + £ t 0 , such that Below, we will sometimes denote a Ricci soliton structure by (Mn,g,X), in order to emphasize the underlying manifold, e.g. when M is a Lie group.
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