All examples about geometric flow equations given in Wikipedia's Geometric flow article are first order in time derivative. Would it make sense to have a geometric flow equation which was second order in time derivative and is there examples of those? If we replace the first order time derivative with second order time derivative in some geometric flow equation (e.g Ricci flow) then does the new equation make sense?

  • $\begingroup$ Do you have a specific model in mind? $\endgroup$ – András Bátkai Sep 11 at 11:59
  • $\begingroup$ No, I'm just curious whether such equations have been studied. $\endgroup$ – Professor Kirby Sep 11 at 12:36
  • $\begingroup$ Indeed, the analog of Ricci flow with a second order time derivative has been studied by D-X Kong and K Liu, see e.g. arxiv.org/pdf/math/0610256.pdf arxiv.org/pdf/0709.1607.pdf and more papers available on Mathscinet $\endgroup$ – YangMills Sep 14 at 13:46

Geometric flows are modeled on fundamental flow equations, namely the heat and wave equations. The heat equation, which uses only a first order derivative in time, diffuses the function, so when $t \rightarrow \infty$, the solution not only converges to a limit, usually zero, but its derivatives all converge to zero. On the other hand, solutions to the wave equation, which uses a second order derivative in time, are oscillatory and do not ever settle down to a stable solution.

In principle, one can study geometric flows using either approach. However, the most common goal is to try to prove that a space satisfying certain geometric properties (such as positive Ricci curvature) also has a unique canonical structure (such as an Einstein metric). The idea is to use a geometric heat flow and try to show that no matter what the starting point is, it always converges to the canonical structure. This is exactly what Hamilton did with the Ricci flow, and this sparked a widespread effort to do this in other settings. That this might work is based on how a solution to the standard heat equation behaves.

The only reason why geometric wave equations have not been studied as intensely is that no one has succeeded yet in showing how it might be used to solve a geometric question. Otherwise, there's not a lot of motivation for geometric analysts to study the equation. The one notable exception is Einstein's equations on a Lorentzian manifold, which is a geometric wave equation.

It is possible that even a geometric wave equation has solutions that converge to a limit. However, the oscillatory nature of the solution makes it much harder to work with than a solution to the geometric heat flow. This, by the way, is why papers on the Einstein equations are often hundreds of pages long.


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