The so-called sausage metrics have the form
$$ g(t) = \frac{-4\sinh(2t)(dx^{2} + dy^{2})}{1 + 2\cosh(2t) r^{2} + r^{4}}$$
in which $z = x + iy$ is a standard coordinate on the complement of a point in the complex projective line, and $r = \sqrt{x^{2} + y^{2}}$. There is an obvious rotational symmetry. The scalar curvature $R(t)$ of $g(t)$ satisfies the bounds
$$\frac{-2}{\sinh(2t)} = \min_{S^{2}}R(t) \leq R(t) \leq \max_{S^{2}}R(t) = -2\coth(2t).$$
The reason for me writing this is that these metrics which are rotationally symmetric metric on the sphere, the Ricci flow can be rewritten as the logarithmic diffusion equation $u_{t} = (\log u)_{zz}$ (not the same $z$ as above).
In the Ricci flow literature they are usually called the Rosenau metrics or the King-Rosenau metrics. However, Fateev-Onofri-Zamolodchikov found this metric, which they called the sausage metric, earlier, in the context of studying the renormalization group flow for a two-dimensional sigma model, in their paper , Integrable deformations of the O(3) sigma model. The sausage model. This Ricci flow is defined as the one-loop renormalization group flow.
In the linked paper, the authors begin by setting up a non-linear sigma model in two-dimensional space-time in the context of a field theory as a continuous model of two-dimensional spin systems as well as in relation to string theory.
The general two-dimensional sigma model (SM) is defined through the action
$$\mathcal{A}(G) = \frac{1}{2}\displaystyle \int G_{ij}(X)\partial_\mu X^i \partial_\mu X^j \mathrm{d}^2x + \ldots$$
where coordinates $x^{\mu},\, \mu = 1, 2$ span a two-dimensional flat space-time, while the fields $X^{i},\, i = 1,2,.. . , d$ are coordinates in a $d$-dimensional Riemann manifold. $M$ is called the target space. The symmetric matrix $G_{ij}(X)$ is the corresponding metric tensor.
If the curvature of $G_{ij}$ is small the action in the above model and is perturbatively renormalizable and one can use the following one-loop renormalization group (RG) evolution equation
$$\frac{\mathrm{d}}{\mathrm{d}t}G_{ij} = -\frac{1}{2}R_{ij} + \mathcal{O}(R^2)$$
where t is the RG “time” (the logarithm of scale) and $R_{ij}$ is the Ricci tensor of
G.
