[en] Fateev’s sausage sigma models in two and three dimensions are known to be integrable. We study their stability under renormalization group (RG) flow in the target space by using results from the mathematics of Ricci flow. We show that the three-dimensional sausage is unstable, whereas the two-dimensional sausage appears to be stable at least at leading order as it approaches the sphere. We speculate that the stability results obtained are linked to the classification of ancient solutions to Ricci flow (i.e., sigma models that are nonperturbative in the infrared regime) in two and three dimensions. We also describe a class of perturbations of the three-dimensional sausage (with the same continuous symmetries) which remarkably decouple. This indicates that there could be a new solution to RG flow, which is described at least perturbatively as a deformation of the sausage. (paper)