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

TL;DR

This paper proves vanishing and finiteness theorems for p-harmonic 1-forms on submanifolds under certain curvature conditions, extending results to harmonic functions and maps.

Contribution

It establishes new vanishing and finiteness theorems for p-harmonic 1-forms based on BiRic curvature bounds, generalizing previous results.

Findings

Vanishing theorem for p-harmonic 1-forms under BiRic curvature conditions.

Finiteness theorem for p-harmonic 1-forms with lower curvature bounds.

Extension of results to p-harmonic functions and maps.

Abstract

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{BiRic}}^a \geq \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{BiRic}}^a \geq -k^2 $.