Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Properties of the solutions of the conjugate heat equation   In this paper we consider the class $\mathcal{A}$ of those solutions $u(x,t)$
to the conjugate heat equation $\frac{d}{dt}u = -\Delta u + Ru$ on compact
K\"ahler manifolds $M$ with $c_1 > 0$ (where $g(t)$ changes by the unnormalized
K\"ahler Ricci flow, blowing up at $T < \infty$), which satisfy Perelman's
differential Harnack inequality on $[0,T)$. We show $\mathcal{A}$ is nonempty.
If $|\ric(g(t))| \le \frac{C}{T-t}$, which is alaways true if we have type I
singularity, we prove the solution $u(x,t)$ satisfies the elliptic type Harnack
inequlity, with the constants that are uniform in time. If the flow $g(t)$ has
a type I singularity at $T$, then $\mathcal{A}$ has excatly one element.

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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
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The categories of this paper are: econ
10
The categories of this paper are: q-bio
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The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: stat
31
The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: cond-mat
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: physics