The stability for F-Yang-Mills functional on CP^n

TL;DR

This paper investigates the stability properties of generalized F-Yang-Mills connections on complex projective spaces, establishing conditions under which stable connections are flat or have specific curvature structures, extending classical Yang-Mills theory.

Contribution

It provides new stability criteria for F-Yang-Mills connections on ^n, including flatness conditions and curvature structure results, generalizing previous Yang-Mills stability results.

Findings

Weakly stable F-Yang-Mills connections are flat under certain conditions.

Characterization of curvature structures for stable connections when equality holds.

A gap theorem for F-Yang-Mills connections on ^n.

Abstract

In this paper, we study the critical points of $F$-Yang-Mills functional on $\mathbb{C}P^n$, which are called $F$-Yang-Mills connections, which is a generalization of Yang-Mills connections. We prove that if $(2+\frac4n)F''(x)x+(n+1)F'(x)<0$, then the weakly stable $F$-Yang-Mills connection on $\mathbb{C}P^n$ must be flat. Moreover, if $(2+\frac4n)F''(x)x+(n+1)F'(x)=0$, we obtain the structure of curvatures corresponding to weakly stable connections. We also show a gap theorem for $F$-Yang-Mills connections on $\mathbb{C}P^n$. Our approach is inspired by Lawson-Simons' study of Yang-Mills stability on spheres.