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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)

$\frac{∂^2}{∂t^2}g_{ij}(t)=-2R_{ij}$

Do 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?

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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. –  Deane Yang Jul 2 '12 at 14:25
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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. –  Deane Yang Jul 2 '12 at 14:42
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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. –  Deane Yang Jul 2 '12 at 18:33
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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. –  Deane Yang Jul 3 '12 at 13:47
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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. –  YangMills Jul 16 '12 at 22:09

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