Instruct: Identify the main and secondary category of Arxiv papers based on the titles
Query: A sharp Sobolev inequality on Riemannian manifolds
17
The categories of this paper are: q-fin
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex