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

Abstract

In this paper, we will show vanishing theorem of p harmonic 1 form on submanifold M in M whose BiRic curvature satisfying BiRic a ≥ Φ 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 M whose BiRic curvature satisfying BiRic a ≥ −k 2 .