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AbstractTwo vanishing theorems for harmonic map andL2harmonic 1-form on complete noncompact manifolds are proved under certain geometric assumptions, which generalize results of , , , , and . As applications, we improve some main results in , , , , , , , , and .doi:10.1017/s0027763000010515 fatcat:3ejapmr7ovfihhcmkojuok5a2q