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?



Ricci flow and isometry group
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 $(X,d)$. 


