Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Second Eigenvalue of Paneitz Operators and Mean Curvature   For $n\geq 7$, we give the optimal estimate for the second eigenvalue of
Paneitz operators for compact $n$-dimensional submanifolds in an
$(n+p)$-dimensional space form.

28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph