Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Stability of K\"ahler-Ricci flow in the space of K\"ahler metrics   In this paper, we prove that on a Fano manifold $M$ which admits a
K\"ahler-Ricci soliton $(\om,X)$, if the initial K\"ahler metric
$\om_{\vphi_0}$ is close to $\om$ in some weak sense, then the weak
K\"ahler-Ricci flow exists globally and converges in Cheeger-Gromov sense.
Moreover, if $\vphi_0$ is also $K_X$-invariant, then the weak modified
K\"ahler-Ricci flow converges exponentially to a unique K\"ahler-Ricci soliton
nearby. Especially, if the Futaki invariant vanishes, we may delete the
$K_X$-invariant assumption. The methods based on the metric geometry of the
space of the K\"ahler metrics are potentially applicable to other stability
problem of geometric flow near a critical metric.

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The categories of this paper are: chem-ph
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: physics
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The categories of this paper are: astro-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph