# Intuitive interpretation of Ricci Flow

What is the best way to interpret, explain or somehow visualize the basic idea behind formal definition of Ricci Flow?

I am familiar with the hackneyed expressions like

"Ricci Flow is a non-linear analogue for the heat equation which smoothens metric",

or

"Ricci Flow describes the deformation of the Riemannian metric tensor on manifold".

However, I was looking for something similar to the balloon-under-pressure interpretation of the mean curvature flow, in particular of the surface tension flow:

Ultimately, I am looking for something that would make Ricci Flow concept clear for undergraduate students without diving too deep into technicalities of Ricci tensor and volume forms.

• In general I don't think you can really get a better answer than "it's a co-ordinate independent heat equation for the metric" without getting technical. You can visualize Ricci flow of surfaces of revolution (see projecteuclid.org/download/pdf_1/euclid.em/1128371754) but the intrinsic nature of the flow makes it very hard to visualize in general. – Anthony Carapetis Apr 26 '15 at 9:49
• Hi, where does the nice image of the surface tension flow come from? Did you make it? – unity Jul 6 '18 at 18:56

The paper in the comment above gives some cool visualizations, but I thought I would mention some easy-to-understand computational interpretations, which can help give intuition (at least to me), even if they are not exactly physical in nature.

Shape Analysis in 3D

See Zeng et al, Ricci Flow for 3D Shape Analysis. Basically, the Ricci flow deforms an arbitrary 3d shape into one of $\mathbb{S}^2$, $\mathbb{H}^2$, or $\mathbb{R}^2$, depending on the shape's topology (and of course the resulting scalar Ricci curvature). The author's note that (in the 3D case) for a closed surface, if the total surface area is preserved during the flow, the Ricci flow converges to a metric with constant Gaussian curvature everywhere. Classic differential geometry is probably more intuitive.

Image Processing

See Appleboim et al, Ricci Curvature and Flow for Image Denoising and Super-Resolution. In it, the authors treat greyscale images as Riemannian manifolds and note that (with their particular construction of the image manifold) running the Ricci flow is essentially analogous to evolving (diffusing) the gradient of the image, rather than the pixel values.

Ito Diffusion Behaviour The behaviour of Brownian motion on manifolds is such that negative scalar curvature is known to accelerate diffusion outward from the starting point (while positive slows it down). See e.g. Debbasch et al, Diffusion Processes on Manifolds. Maybe with this in mind, one can imagine Ricci flow to be "smoothing out" the behaviour of a diffusing particle on the manifold, in a way.

We do not know asymptotic at infinity of $Ric +\nabla^2 f=Cg,\ C<0$ where $M$ is noncompact manifold. Then $g(t)=c(t)\phi(t)^\ast g(0)$ where $\phi$ is flow of $\nabla f$. Here $g(t)$ converges to flat when ${\rm dim}\ M=2$.