Instability of solutions to the Ginzburg-Landau equation on $S^n$ and ${\mathbb{C}\mathbb{P}}^n$

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 $n\ge 2$ and ${\mathbb{C}\mathbb{P}}^n$ with $n\ge 1$, stable critical points must be constants. In addition, for GL critical points on $S^n$ for $n\ge 3$ 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 $n\ge 4$ 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.