Suppose $\omega$ defines a Kähler metric on a non-compact complex manifold. Does the Kähler-Ricci flow equation always have a solution (for small $t$)?
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2$\begingroup$ Without extra assumptions, the answer is no even on $\mathbb{C}^n$. $\endgroup$– YangMillsCommented Jan 10, 2015 at 21:07
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$\begingroup$ Thanks for the answer! Then on C^n what condition will ensure it has a solution? Any reference? Much appreciated! $\endgroup$– Mohawk RiverCommented Jan 11, 2015 at 2:02
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$\begingroup$ As you can see, I am from a different math background. $\endgroup$– Mohawk RiverCommented Jan 11, 2015 at 2:03
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$\begingroup$ math.ucsd.edu/~lni/academic/Ni.pdf looks like a nice survey $\endgroup$– Deane YangCommented Jan 11, 2015 at 12:32
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1$\begingroup$ If the initial metric can be extended to a complete metric on $\mathbb{C}^n$ or $\mathbb{P}^n$ with the right assumptions, then there is a solution but no uniqueness. $\endgroup$– Deane YangCommented Jan 11, 2015 at 15:32
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1 Answer
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Lets start with a definition:
Definition of conic metric on complement of divisor: If we take $X^{reg}=X\setminus D$, where D is a divisor with normal crossings. Endow $X^{reg}$ with a complete Kähler metric which has a type of singularities normal to each component of $D$; in local coordinates, if $D = (z_1,...,z_k)$, the conic metric $\omega$ is quasi-isometric with the following model, $$\sum_{i=1}^k\frac{dz_i\wedge d\bar z_i}{|z_i|^2}+\sum_{i=k+1}^n dz_i\wedge d\bar z_i$$
If $D$ be a simple normal crossing divisor on Kähler variety $X$ with some type of singularities, like conic, cusp,...., then the Kähler Ricci flow
$$\frac{\partial \omega}{\partial t}=-Ric(\omega)+\lambda\omega$$
has solution on non-compact and quasi projective, open variety $X\setminus D$ when $\lambda=0, -1$
Reference: Paper of Unnormalize conical Kähler-Ricci flow Liangming Shen http://arxiv.org/abs/1411.7284
- $C^{2,α}$-estimate for conical Kähler-Ricci flow http://arxiv.org/abs/1412.2420