Vanishing theorem and finiteness theorem for $p$-harmonic $1$ form

In this paper, we will show vanishing theorem of $p$ harmonic $1$ form on submanifold $M$ in $\bar{M}$ whose BiRic curvature satisfying ${\overline{\mathrm{B}\mathrm{i}\mathrm{R}\mathrm{i}\mathrm{c}}}^a\ge {\mathrm{\Phi }}_a(H,S)$. As an corollary, we can get the corresponding theorem for $p$ harmonic function and $p$ harmonic map. We also investigate the finiteness problem of $p$ harmonic $1$ form on submanifold $M$ in $\bar{M}$ whose BiRic curvature satisfying ${\overline{\mathrm{B}\mathrm{i}\mathrm{R}\mathrm{i}\mathrm{c}}}^a\ge -k^2$.