After reading a bit more about Perelman's entropy and gradient solitons, I came up with a hunch, which I must test. Non-singular solitons can be regarded as critical points of Perelman's entropy, or fixed points of the Ricci flow characterized by $$R_{ij} + \lambda g_{ij} + \nabla _i \nabla _j f = 0 .$$ My question is about the function $f$ when it does exist. Is there any flexibility in the choice of $f$ for a particular soliton? That is to say, at any fixed time, is $f$ always uniquely determined? Please, a couple or more of explicit solitons with the accompanying $f$'s would be highly appreciated so as to run some tests with the F- and W- functionals.
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Even if this is an old question, it may be useful, if not for the OP, for somebody that finds it.
Assume we have a Ricci soliton $(M,g,f)$ with constant $\lambda \in \mathbb R$ (i.e., the equation in the OP is satisfied). For such fixed $\lambda$, on a compact manifold, $f$ is the only function satisfying such equation up to constants: if another function does, their difference has zero hessian. When the manifold is non compact, this might be more involved (you would need to study the space of "linear" functions, i.e. those with zero hessian).
More interestingly, when you allow $\lambda$ to vary, this is not the case anymore. For example $\mathbb R^n$ can be seen as a shrinking, steady, or expanding soliton: see for example Topping's lecture notes, pag. 11. In particular it is a rigid case for all of Perelman's monotonicities, choosing the right $f$. However, if $M$ is compact, this cannot happen: tracing and integrating the soliton equation, one can see that $\lambda$ is determined by $g$ alone.
