Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Generic existence of multiplicity-1 minmax minimal hypersurfaces via
  Allen--Cahn   In Guaraco's 2018 work a new proof was given of the existence of a closed
minimal hypersurface in a compact Riemannian manifold $N^{n+1}$ with $n\geq 2$.
This was achieved by employing an Allen--Cahn approximation scheme and a
one-parameter minmax for the Allen--Cahn energy (relying on works by
Hutchinson, Tonegawa, Wickramasekera to pass to the limit as the Allen-Cahn
parameter tends to $0$). The minimal hypersurface obtained may a priori carry a
locally constant integer multiplicity. Here we consider a minmax construction
that is a modification of the one in the aforementioned work, by allowing an
initial freedom on the choice of the valley points between which the mountain
pass construction is carried out, and then optimising over said choice. We
prove that, when $2\leq n\leq 6$ and the metric is bumpy, this minmax leads to
a (smooth closed) minimal hypersurface with multiplicity $1$. (When $n=2$ this
conclusion also follows from Chodosh--Mantoulidis's recent work.) As immediate
corollary we obtain that every compact Riemannian manifold of dimension $n+1$,
$2\leq n\leq 6$, endowed with a bumpy metric, admits a two-sided smooth closed
minimal hypersurface (this existence conclusion also follows from Zhou's recent
result for minmax constructions via Almgren--Pitts theory).

14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics