Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complete surfaces with positive extrinsic curvature in product spaces   We prove that every complete connected immersed surface with positive
extrinsic curvature $K$ in $H^2\times R$ must be properly embedded,
homeomorphic to a sphere or a plane and, in the latter case, study the behavior
of the end. Then, we focus our attention on surfaces with positive constant
extrinsic curvature ($K-$surfaces). We establish that the only complete
$K-$surfaces in $S^2\times R$ and $H^2\times R$ are rotational spheres. Here
are the key steps to achieve this. First height estimates for compact
$K-$surfaces in a general ambient space $M^2\times R$ with boundary in a slice
are obtained. Then distance estimates for compact $K-$surfaces (and
H-$surfaces) in $H^2\times R$ with boundary on a vertical plane are obtained.
Finally we construct a quadratic form with isolated zeroes of negative index.

3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
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The categories of this paper are: cs
29
The categories of this paper are: funct-an
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The categories of this paper are: eess