A splitting theorem for the weighted measure

Let M be an n -dimensional complete noncompact Riemannian manifold, h be a smooth function on M and d μ = e h d V be the weighted measure. In this article, we prove that when the spectrum of the weighted Laplacian has a positive lower bound λ 1 ( M ) > 0 and the m ( m > n )-dimensional Bakry-Émery curvature is bounded from below by , then M splits isometrically as R × N whenever it has two ends with infinite weighted volume, here N is an ( n − 1)-dimensional compact manifold.