Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Min-max theory for free boundary minimal hypersurfaces II -- General
  Morse index bounds and applications   For any smooth Riemannian metric on an $(n+1)$-dimensional compact manifold
with boundary $(M,\partial M)$ where $3\leq (n+1)\leq 7$, we establish general
upper bounds for the Morse index of free boundary minimal hypersurfaces
produced by min-max theory in the Almgren-Pitts setting. We apply our Morse
index estimates to prove that for almost every (in the $C^\infty$ Baire sense)
Riemannan metric, the union of all compact, properly embedded free boundary
minimal hypersurfaces is dense in $M$. If $\partial M$ is further assumed to
have a strictly mean convex point, we show the existence of infinitely many
compact, properly embedded free boundary minimal hypersurfaces whose boundaries
are non-empty. Our results prove a conjecture of Yau for generic metrics in the
free boundary setting.

32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex