Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Normalized Ricci Flow on Four-Manifolds and Exotic Smooth Structures   In this article, we shall investigate the relationship between the existence
or non-existence of non-singular solutions to the normalized Ricci flow and
smooth structures on closed 4-manifolds, where non-singular solutions to the
normalized Ricci flow are solutions which exist for all time $t \in [0,
\infty)$ with uniformly bounded sectional curvature. In dimension 4, there
exist many compact topological manifolds admitting distinct smooth structures,
i.e., exotic smooth structures. Interestingly, in this article, the difference
between existence and non-existence of non-singular solutions to the normalized
Ricci flow on 4-manifolds turns out to strictly depend on the choice of smooth
structure. In fact, we shall prove that, for every natural number $\ell$, there
exists a compact topological 4-manifold $X_{\ell}$ which admits smooth
structures for which non-singular solutions of the normalized Ricci flow exist,
but also admits smooth structures for which no non-singular solution of the
normalized Ricci flow exists. Hence, in dimension 4, smooth structures become
definite obstructions to the existence of non-singular solutions to the
normalized Ricci flow.

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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
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The categories of this paper are: comp-gas
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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The categories of this paper are: supr-con
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The categories of this paper are: math-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: gr-qc