Instability of solutions to the Ginzburg-Landau equation on Sn and CPn

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 >= 2 and CPn with n >= 1, stable critical points must be constants. In addition, for GL critical points on S-n 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 S-n with n >= 4 there are no stable critical points unless the line bundle is isomorphic to S-n x C, in which case the only stable critical points are the trivial ones. Our methods come from the work of Lawson-Simons. (C) 2020 Elsevier Inc. All rights reserved.