Vanishing theorems on complete manifolds with weighted Poincaré inequality and applications

Abstract

Two vanishing theorems for harmonic map and $L^2$ harmonic $1$-form on complete noncompact manifolds are proved under certain geometric assumptions, which generalize results of [13], [15], [18], [19], and [20]. As applications, we improve some main results in [2], [4], [6], [9], [12], [20], [22], [24], and [25].