Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Instability of solutions to the Ginzburg-Landau equation on $S^{n}$ and
  $\mathbb{CP}^{n}$   We study critical points of the Ginzburg-Landau (GL) functional and the
abelian Yang-Mills-Higgs (YMH) functional on the sphere and the complex
projective space, both equipped with the standard metrics. For the GL
functional we prove that on $S^{n}$ with $n \geq 2$ and $\mathbb{CP}^{n}$ with
$n \geq 1$, stable critical points must be constants. In addition, for GL
critical points on $S^{n}$ for $n \geq 3$ we obtain a lower bound on the Morse
index under suitable assumptions. On the other hand, for the abelian YMH
functional we prove that on $S^{n}$ with $n \geq 4$ there are no stable
critical points unless the line bundle is isomorphic to $S^n \times
\mathbb{C}$, in which case the only stable critical points are the trivial
ones. Our methods come from the work of Lawson--Simons.

9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int