Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Classification results for expanding and shrinking gradient
  K\"ahler-Ricci solitons   We first show that a K\"ahler cone appears as the tangent cone of a complete
expanding gradient K\"ahler-Ricci soliton with quadratic curvature decay with
derivatives if and only if it has a smooth canonical model (on which the
soliton lives). This allows us to classify two-dimensional complete expanding
gradient K\"ahler-Ricci solitons with quadratic curvature decay with
derivatives. We then show that any two-dimensional complete shrinking gradient
K\"ahler-Ricci soliton whose scalar curvature tends to zero at infinity is, up
to pullback by an element of $GL(2,\,\mathbb{C})$, either the flat Gaussian
shrinking soliton on $\mathbb{C}^{2}$ or the $U(2)$-invariant shrinking
gradient K\"ahler-Ricci soliton of Feldman-Ilmanen-Knopf on the blowup of
$\mathbb{C}^{2}$ at one point. Finally, we show that up to pullback by an
element of $GL(n,\,\mathbb{C})$, the only complete shrinking gradient
K\"ahler-Ricci soliton with bounded Ricci curvature on $\mathbb{C}^{n}$ is the
flat Gaussian shrinking soliton and on the total space of
$\mathcal{O}(-k)\to\mathbb{P}^{n-1}$ for $0<k<n$ is the $U(n)$-invariant
example of Feldman-Ilmanen-Knopf. In the course of the proof, we establish the
uniqueness of the soliton vector field of a complete shrinking gradient
K\"ahler-Ricci soliton with bounded Ricci curvature in the Lie algebra of a
torus. A key tool used to achieve this result is the Duistermaat-Heckman
theorem from symplectic geometry. This provides the first step towards
understanding the relationship between complete shrinking gradient
K\"ahler-Ricci solitons and algebraic geometry.

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The categories of this paper are: q-bio
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: stat
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The categories of this paper are: supr-con
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: econ