# Reverse Ricci Flow and Longtime Existence

The usual Ricci flow and normalized Ricci flow for surfaces are $$\partial_t g = -2Kg$$ and $$\partial_t g = -2Kg + 2sg,$$ where $K$ is the Gaussian curvature and $s$ is its average. The latter equation can be solved on the interval $[0,\infty)$ and converges to a metric of constant Gaussian curvature.

Now consider the reverse flow $$\partial_t = 2Kg - 2sg,$$ which should function as some sort of reverse flow. What can you say about its long-time existence?

I had a discussion about this with someone, who argued that there should be no time-reversal of that flow as the flow would not know "where to go" if you start in, say, a sphere of constant curvature (because of the above convergence result; this flow should move the metric away from a constant curvature metric).

But clearly, at least for round spheres the above equation has a unique solution, as $\partial_t g = 0$. So why can't a round sphere, for instance, be a repelling fix point of the flow... Writing this, I realize that this flow, if it existed, would have one crucial disadvantage: $g(t)$ would not depend continuously on $g$ in the $C^k$-topology.

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One remark, if the metric $g$ is not analytic, then there is no time-reversal, by a result of Bando (the metric becomes instantaneously analytic under Ricci flow). –  Ian Agol Sep 28 at 18:24
Even if the metric is analytic, there may still be no time-reversal. Even the classical linear heat equation $u_t = u_{xx}$ on the circle (i.e., periodic in $x$) does not have time reversal for most analytic initial data. Only if the Fourier coefficients decay extremely rapidly, more rapidly than is required for analyticity, will there be even short-time existence for the time-reversed flow. –  Robert Bryant Sep 28 at 21:33
However, one can show that if such backward solution exist, it is unique see [arxiv.org/abs/0906.4920] –  Thomas Richard Oct 6 at 7:46
See the related MSE question, "Reversing the Ricci flow" –  Joseph O'Rourke Nov 29 at 3:37
As Ian Agol wrote, Shigetoshi Bando, by way of summable derivative of curvature estimates of Bernstein type, proved that for a solution $(M,g(t)),$ $t\in(0,T)$, to the Ricci flow on a closed manifold, at each $t$ the metric $g(t)$ is real analytic (in space). These summable estimates and the consequent real analyticity result can be localized; see arXiv.1111.0355 by Brett Kotschwar. Also, in arXiv.1210.3083, Kotschwar proves that $g(t)$ is real analytic in both space and time. This involves careful estimates for $t^{k+2\ell}|\nabla^{k}\partial_{t}^{\ell}\operatorname{Rm}|^{2}$. As Thomas Richard wrote, backwards uniqueness was obtained by Kotschwar in arXiv.0906.4920.
The following somewhat unrelated facts come to mind regarding long time existence backwards. By this I mean an ancient solution to the Ricci flow on a closed surface. Either it is flat or $R>0$. In the latter case, if it is also Type I, Hamilton proved that the solution must be a round shrinking sphere or its $\mathbb{Z}_2$ quotient. If it is Type II, then Daskalopoulos, Hamilton and Sesum proved that it must be the King-Rosenau solution or its $\mathbb{Z}_2$ quotient. Note that the King-Rosenau solution is rotationally symmetric and invariant under a reflection. Since (loosely speaking) its backwards limits are two opposing cigar steady solitons, one may think of the King-Rosenau solution as a heteroclinic orbit joining two cigars (fixed points modulo conformal diffeomorphisms) to the round sphere (a fixed point modulo scaling).