  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.
