# Reversing the Ricci flow

Suppose $S$ is a closed, oriented surface (2-manifold) embedded in $\mathbb{R}^3$, which inherits the metric from $\mathbb{R}^3$, so that distances are measured by shortest paths on the surface. If it is at least crudely accurate to say that Ricci flow smooths out the metric/curvature so that the surface (eventually) evolves to a sphere, is there a sense in which "reverse Ricci flow" concentrates curvature, in some sense sharpens the curvature, and perhaps partitions $S$ into distinct regions?

This is a naive question, for which I apologize in advance; it could be complete nonsense. What I mean by reverse Ricci flow (my own terminology; perhaps there is standard terminology?), could be simply changing the sign in Hamilton's equation:

$$\frac{\partial g}{\partial t} \ = \ 2 \ {\bf Ric}(g) \;.$$

What I mean by "sharpens the curvature" is something akin to image-processing operators which enhance the boundaries between regions to segment an image (edge detection). I am imagining segmenting a surface $S$ by reverse Ricci flow.

One problem I can foresee is that is that reversing the heat equation is inherently unstable, and maybe the same is true here. But perhaps in the specific situation of $S \subset \mathbb{R}^3$ inheriting the Euclidean metric, the instabilities are not as severe as they might be for arbitrary Riemannian manifolds.

All this is speculation on my part. Reality checks, references, or further speculation—all welcomed!

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What you are probably after is something like mean curvature flow. This evolution equation does concentrate curvature -- a good reference for this happening with convex initial hypersurfaces is Huisken's seminal 1984 paper in JDG. More recently, people have worked on surgery for the mean curvature flow, which does break the hypersurface into distinct regions where the curvature concentrates into singularities.

For your specific question, it isn't possible. Uniqueness in backward time is rubbish: the PDE is not parabolic. There is a clear and intuitive reason for this. The Ricci flow evolves geometries toward a model state: on a sphere with positive curvature, toward the geometry of a sphere. Now think about this for a second. All positive curvature geometries evolve toward one geometry -- as you wrote, the effect of the Ricci flow is to `smooth out' regions of curvature. Now if I begin at the goal model geometry, and flow in the backward time direction, how can I possibly know which less-ideal-but-positively-curved geometry to pick?

This is the basic problem with equations which aren't parabolic. In general, to reverse time is a very special and tricky thing and you need a lot more information to do it with any meaning. In this sense, I'd venture that the Ricci flow isn't the tool you're looking for.

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This is very useful intuition, Glen---Thanks! –  Joseph O'Rourke Oct 7 '12 at 2:01

In http://en.wikipedia.org/wiki/Ricci_flow#Relation_to_diffusion you will find an explicit computation of the form of the equation on a 2-dimensional manifold. The equation is, up to trivial manipulations, the usual heat equation in the plane. This is a non-reversible evolution equation.

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I thought that the heat equation, roughly, had unique backward time solutions? I.e. if two initial conditions give the same time evolution then they are the same. (I'm not an expert.) –  Grumpy Parsnip Oct 15 '11 at 2:33
I know little about this, but there is an Osher-Rudin approach to solving a time-reversed heat equation to effect image deblurring... –  Joseph O'Rourke Oct 15 '11 at 2:46
@Jim, if I remember correctly, there is uniqueness but not existence in general, no? –  Mariano Suárez-Alvarez Oct 15 '11 at 2:48
Hmm. I probably shouldn't stick my nose into an area of which I have so little knowledge, but I agree with that. So I guess the question includes: "Under what conditions can the flow be reversed, and how far can it be reversed before hitting some kind of singularity?" –  Grumpy Parsnip Oct 15 '11 at 2:53
@GrumpyParsnip: time-reversing heat flow equation is impossible, provided that heat diffusion process by definition generate entropy and thus the information is lost. This is not true for most of the differential equations, notably in solid dynamics, where intial conditions and equation uniquely define all time positions. The reason it is true for heat diffusion equations is that it not with a set of point in canonical corrdinates but a volume. However, because of convergence of solutions of diffusion equation, this volume diverges if time is reversed. –  Andrei Kucharavy Apr 19 '14 at 20:51