Could someone please give me a reference where I can find a complete proof of the result Ricci flow preserves holonomy? Is there any way to prove that Ricci flow preserves Kahler condition without using that theorem?
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3$\begingroup$ Do you find arxiv.org/abs/1105.3722 and references therein insufficient? $\endgroup$– Igor BelegradekCommented Aug 8, 2016 at 12:23
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1$\begingroup$ Let $H$ be the holonomy group and $\mathfrak{h}$ be the corresponding subalgebra in $\mathfrak{so}(n)$. The algebra $\mathfrak{so}(n)$ can be identified with the space of bivector $\Lambda^2(T)$. Note that $\mathfrak{h}$ forms a parallel distribution in and the curvature operator has values in $\mathfrak{h}$. It remains to apply the formula for curvature evolution. $\endgroup$– Anton PetruninCommented Aug 8, 2016 at 12:24
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7$\begingroup$ For your last question, you can also prove directly that the Kahler-Ricci flow has a solution for short time, by writing it as a parabolic scalar complex Monge-Ampere equation. By construction, the solution is Kahler as long as it exists. But it also solves the Riemannian Ricci flow, so by uniqueness of Ricci flow solutions you conclude that the Ricci flow preserves Kahler. $\endgroup$– YangMillsCommented Aug 8, 2016 at 15:34
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Actually, the only nontrivial case is the Kahler case:
If you put on the assumptions you need in order for the Ricci flow to be uniquely defined, then by passing to a cover, you can assume that $(M,g)$ is the direct product of $(M_i,g_i)$, each factor having irreducibly acting holonomy (de Rham Splitting Theorem). Thus, you are reduced to the case that the holonomy acts irreducibly.
Assuming that $(M,g)$ has irreducible holonomy, then, by the Berger Holonomy Classification Theorem, there are only three possibilities: (i) the holonomy of $(M^n,g)$ is $\mathrm{SO}(n)$, (ii) the holonomy of $(M^{2n},g)$ is $\mathrm{U}(n)\subset\mathrm{SO}(2n)$, or (iii) the metric $g$ is Einstein (or even Ricci-flat). In cases (i) and (iii), there is nothing to prove. The case (ii) is the Kähler case, and, as has been pointed out in the comments, you just need to check that the Ricci-flow in the Kähler case can be reformulated as a flow on the corresponding (1,1)-form, i.e., the underlying complex structure does not change. Then uniqueness does the rest.
answered Aug 13, 2016 at 13:45