How we can prove the short time existence for Kahler Ricci flow on compact manifolds. I know that by linearization and Detork's trick we can prove the short time existence for Ricci flow, but for proving on Kahler Ricci flow we have to use of second order parabolic equations. Of course on noncompact manifolds I have seen a nice proof but In this question I mean on compact manifolds.
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First, the Kahler-Ricci flow is the Ricci flow applied to a Kahler metric (which is a Riemannian metric), so any theorem and proof for local time existence of the Ricci flow applies to the Kahler-Ricci flow.
Second, as far as I know, any proof for short time existence that works for noncompact manifolds will work without any change for a compact manifold (without boundary).
Third, as Otis has pointed out, the Kahler-Ricci flow can be written as an honest nonlinear parabolic PDE for the potential. If you believe that a linear parabolic PDE can be solved, then the nonlinear theorem is easily proved using energy integral estimates and the implicit function theorem.
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