Vanishing theorems for f-harmonic forms on smooth metric measure spaces. (October 2017) - Han, Yingbo; Lin, Hezi
Abstract: 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 L 2 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 L q f -harmonic 1 -forms.