We know

  • Laplace equation (elliptic equations) $ Δ u = 0$
  • Heat equation (parabolic equations) $u_t − Δu = 0$
  • Wave equation (hyperbolic equations) $u_{tt} − Δu = 0$

we have - Hyperbolic geometric flow (hyperbolic equations)


Does any body know

The short time existence for heat equation Hyperbolic Ricci flow, when our Manifold is non-compact?

and second question:

Hyperbolic Ricci flow is invariant under first Chern class?

  • 1
    $\begingroup$ What's rather important here are your assumptions on the initial data, as well as the behavior of the metric for positive time at spatial infinity. You get to specify not only the initial metric $g(0)$ but also its initial time derivative $\partial_tg(0)$. If you assume that $\partial_tg(0)$ decays fast enough at spatial infinity and require that $partial_tg(t)$ has the same kind of decay (probably expressed in terms of the boundedness of an energy integral), then it should be easy to adapt known techniques for hyperbolic PDE's. $\endgroup$ – Deane Yang Jul 2 '12 at 14:25
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    $\begingroup$ Your second question, I assume, is for a Kahler metric? I don't know anything about this stuff, especially on a non-compact manifold, my guess is that it will be invariant only if $\partial_tg(t)$ decays fast enough at spatial infinity so that the argument (which involves integration by parts) that shows the first Chern class is invariant under the flow on a compact manifold can be used in the non-compact case. $\endgroup$ – Deane Yang Jul 2 '12 at 14:42
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    $\begingroup$ Haskell, thanks for asking but since what I wrote was just a wild guess, I don't have any further explanation for it. Is a similar fact true for the Ricci flow on a noncompact Kahler manifold? If so, there might be a similar proof for the hyperbolic flow. $\endgroup$ – Deane Yang Jul 2 '12 at 18:33
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    $\begingroup$ In fact, I would like to retract my comment above ("Your second question, I assume,...). The more I think about it, the more I realize how little I know about the first Chern class of a noncompact Kahler manifold. $\endgroup$ – Deane Yang Jul 3 '12 at 13:47
  • 3
    $\begingroup$ I think your best bet is to ask De-Xing Kong directly (kong@cms.zju.edu.cn). He is the person who introduced this flow together with Kefeng Liu. It seems to me that the only papers on this topic that have been written so far (about 10 in total) are either by these two people or by their students. As far as I could tell none of those addresses the case of noncompact manifolds. But I second Deane's advice, that probably a lot can be proved if you are willing to make some decay/periodicity assumptions on the initial data. $\endgroup$ – YangMills Jul 16 '12 at 22:09

For the first question take local harmonic coordinates the equation takes the form

$$ \partial^2_{tt} g_{ij} - g^{kl} \partial^2_{kl} g_{ij} = l.o.t. $$

and so using finite speed of propagation local existence holds in the non-compact case provided that the data is not too wild near infinity.

In fact, for any sufficiently smooth initial data, regardless of asymptotic behaviour, standard hyperbolic PDE theory tells you that there exists a sufficiently smooth Cauchy development. The only issue is that what is the "time of existence", as generally for non-compact initial data you may not have a lower bound on the time of existence.

For now assume that your initial noncompact manifold is geodescially complete. In the incomplete case you will have problems coming from the boundary behaviour.

As Deane commented that it suffices that the metric and its first derivative decays in the initial data. But you can get away with even weaker statements. For example, I am pretty sure it suffices to have

  • Uniform lower bound on the radius of injectivity of the initial metric, by rescaling we can take this bound to be 1.
  • Uniform upper bound on the local higher energy: for some $N$ depending on the dimension of your manifold you require $$ \sup_{x\in M} \sup_{0 \leq k \leq N} \| \nabla^k \mathrm{Riem} \|_{L^2(B(x,1))} < \infty $$ and something analogous for the initial velocity: let $h = \partial_t g | _{t = 0}$ be the symmetric two tensor representing the initial velocity, require $$ \sup_{x \in M} \sup_{0 \leq k \leq N-1} \|\nabla^k h\|_{L^2(B(x,1))} < \infty $$

Then using finite speed of propagation and what you already know about in the compact case you get control on the time of existence of the Cauchy developments originating from balls of radius one. Alternatively you can get the same result by modifying one of the many write-ups of the well-known proof in the case of Einstein's equation.

I leave the second question to other experts.


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