Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complete subamanifolds of $\mathbb{R}^{n}$ with finite topology   We show that a complete $m$-dimensional immersed submanifold $M$ of
$\mathbb{R}^{n}$ with $a(M)<1$ is properly immersed and have finite topology,
where $a(M)\in [0,\infty]$ is an scaling invariant number that gives the rate
that the norm of the second fundamental form decays to zero at infinity. The
class of submanifolds $M$ with $a(M)<1$ contains all complete minimal surfaces
in $\mathbb{R}^{n}$ with finite total curvature, all $m$-dimensional minimal
submanifolds $M $ of $ \mathbb{R}^{n}$ with finite total scalar curvature
$\smallint_{M}| \alpha |^{m} dV<\infty $ and all complete 2-dimensional
complete surfaces with $\smallint_{M}| \alpha |^{2} dV<\infty $ and nonpositive
curvature with respect to every normal direction, since $a(M)=0$ for them.

28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
25
The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph