Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Index estimates for free boundary $f$-minimal hypersurfaces   We prove that the index is bounded from below by a linear function of its
first Betti number for any compact free boundary $f$-minimal hypersurface in
certain positively curved weighted manifolds.

13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph