Vanishing theorems for f-harmonic forms on smooth metric measure spaces

In this paper, we first establish a monotonicity formula for vector bundle-valued f-harmonic p-forms on a smooth metric measure space provided 〈∇f,∇r〉 is less that an explicit constant. As applications, we get some vanishing theorems for L2 f-harmonic forms on concrete geometric models. In the second part, for a metric measure space with nonnegative ∞-Bakry-Émery-Ricci curvature and with moderate volume growth, we prove that any bounded f-harmonic 1-form must be parallel. Moreover, some vanishing theorems under nonnegative m-Bakry-Émery-Ricci curvature assumption are also proved. Finally, we consider smooth metric measure spaces with weighted Poincaré inequality and show some vanishing theorems for Lq f-harmonic 1-forms.