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

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 $(2+\frac{4}{n})F^{\mathrm{\prime }\mathrm{\prime }}(x)x+(n+1)F^{\mathrm{\prime }}(x)<0$, then the weakly stable $F$-Yang-Mills connection on $\mathbb{C}{P}^n$ must be flat. Moreover, if $(2+\frac{4}{n})F^{\mathrm{\prime }\mathrm{\prime }}(x)x+(n+1)F^{\mathrm{\prime }}(x)=0$, 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.