Instability and Isolation for Yang-Mills Fields over Compact Submanifold of Sphere

If M is an n-dimensional compact oriented submanifold immersed in a sphere Sn+p, it is proved that, if n 4 + A + 2σ where σ is the square length of the second fundamental form and A is a positive constant depending only on the square length of the second fundamental form and the mean curvature of M, there are no non-trivial weakly stable Yang-Mills fields on M. Thus we generalize a classical result due to Simons that the standard sphere Sn(n 4) is Yang-Mills unstable. It is also shown that there is a gap phenomena for Yang-Mills fields on the compact submanifolds in the sphere.