Ricci flow and isometry group

Question

It is known (via Kotschwar's uniqueness of backwards Ricci flows) that the isometry group of a Riemannian metric remains unchanged under the Ricci flow. But, one can easily observe that it can change at the limit. For example, one can perturb a sphere $S^2$ slightly so that it has no symmetry. As long as the Ricci flow is defined, the resulting surfaces still have trivial isometry groups. But, in the limit, if one is using normalized Ricci flows, the isometry group suddenly jumps to $SO(3)$. My question is, can one say that if the isometry group under normalized Ricci flows changes at the limit, it actually has to get "bigger"? Heuristically, this sounds very plausible, but is it actually true?

Answer 1

I think the phenomenon is much more general: If a sequence of metrics $d_i$ on a compact metric space $X$ converges (pointwise on $X\times X$) to a metric $d$, and if $h$ lies in the intersection of the isometry groups of $(X, d_i)$, then clearly $h$ is an isometry of $(X,d)$.

Answer 2

The answer is yes (I'm assuming you are asking about closed manifolds, non-compactness allows for all sort of crazy things to happen, you can check out the work of Topping and collaborators).

Kotschwar's backwards uniqueness proves that the isometry group cannot increase under the Ricci flow in finite time. The statement that the isometry group cannot decrease is much easier to prove and follows from short time uniqueness of Ricci flow:

If $g=\Phi^*g$, then if $g_t$ is the Ricci flow starting from $g$, note that $\Phi^*g_t$ is also a Ricci flow starting from $g_t$. Thus $g_t = \Phi^*g_t$ (this follows from the short time uniqueness of Ricci flow). Clearly this is the same for the volume preserving flow.

So, what this says is that any isometry that exists at the initial time, is preserved for all time. So, $g_t = \Phi^*g_t$ it must pass to the limit as $t\to\infty$, assuming there is such a limit. Thus $g_\infty = \Phi^*g_\infty$, so the isometry persists in the limit.

On the other hand, the example you give (a non round metric on the $2$-sphere becomes round as $t\to\infty$ for the normalized flow) shows that isometries can appear in the long time limit!

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Added: I would argue that you should not be that surprised by this phenomenon, it appears in parabolic equations all the time. For example, a solution heat equation on an interval (say with Dirichlet boundary conditions) cannot suddenly become $0$ in finite time, by "backwards uniqueness." (In this case, you can probably prove it by hand by using a Fourier series expansion). But in infinite time, all solutions tend to $0$!