L-f(2)-harmonic 1-forms on smooth metric measure spaces with positive lambda(1)(Delta(f))

摘要

In this paper, we study vanishing and splitting results on a complete smooth metric measure space (M-n, g, e(-f) dv) with various negative m-Bakry-Emery Ricci curvature lower bounds in terms of the first eigenvalue lambda(1)(Delta(f)) of the weighted Laplacian Delta(f), i.e., Ric(m, n) >= - a lambda(1)(Delta(f))- b for 0 < a <= m/m - 1, b >= 0. In particular, we consider three main cases for different a and b with or without conditions on lambda(1)(Delta(f)). These results are extensions of Dung and Vieira, and weighted generalizations of Li-Wang, Dung-Sung, and Vieira.