2019/11/11 by Da Rong Cheng, Cheng, Da Rong
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1911.04097
openalex publication_date 2019/11/11 · openalex created_date 2019/11/22 · openalex updated_date 2026/07/28
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 \mathbbCPn 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.