Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Mean curvature flow of area decreasing maps in codimension two   We consider the graphical mean curvature flow of strictly area decreasing
maps $f:M\to N$ between a compact Riemannian manifold $M$ of dimension $m> 1$
and a complete Riemannian surface $N$ of bounded geometry. We prove long-time
existence of the flow and that the strictly area decreasing property is
preserved, when the Ricci curvature $\Ric_M$ on $M$ satisfies the main
condition
  $$\Ric_M(v,v)+\Ric_M(w,w)-\sigma_M(v\wedge w)\,\ge \,{\sup}_N\sigma_N,$$
where $\sigma_M,\sigma_N$ denote the sectional curvatures on $M$ and $N$,
respectively. When $M$ has dimension $2$ or $3$, this condition is equivalent
to $\operatorname{Scal}_M\ge\,\operatorname{Scal}_N$,
  where $\operatorname{Scal}_M, \operatorname{Scal}_N$ denote the scalar
curvatures on $M$ and $N$, respectively. In addition, if $\Ric_M\,\ge
\,{\sup}_N\sigma_N$, then the mean curvature stays uniformly bounded and
uniform $C^1$-bounds for the maps imply uniform $C^k$-bounds, for any $k\ge 1$.
  Under the assumption that the main curvature condition holds, we obtain such
$C^1$-bounds if
  $\Ric_M\ge 0$ and we obtain smooth convergence to a minimal map if
  $\Ric_M\ge\sup\{0,{\sup}_N\sigma_N\}$.
  These results significantly improve known results on the graphical mean
curvature flow in codimension $2$.

31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
16
The categories of this paper are: stat
26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
1
The categories of this paper are: cs
19
The categories of this paper are: eess
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The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas