Instability of solutions to the Ginzburg–Landau equation on Sn and [formula omitted

Abstract

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 Sn with n≥2 and CPn with n≥1, stable critical points must be constants. In addition, for GL critical points on Sn for n≥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 Sn with n≥4 there are no stable critical points unless the line bundle is isomorphic to Sn×C, in which case the only stable critical points are the trivial ones. Our methods come from the work of Lawson–Simons.