  In this paper, we first prove a compactness theorem for the space of closed
embedded $f$-minimal surfaces of fixed topology in a closed three-manifold with
positive Bakry-\'{E}mery Ricci curvature. Then we give a Lichnerowicz type
lower bound of the first eigenvalue of the $f$-Laplacian on compact manifold
with positive $m$-Bakry-\'{E}mery Ricci curvature, and prove that the lower
bound is achieved only if the manifold is isometric to the $n$-shpere, or the
$n$-dimensional hemisphere. Finally, for compact manifold with positive
$m$-Bakry-\'{E}mery Ricci curvature and $f$-mean convex boundary, we prove an
upper bound for the distance function to the boundary, and the upper bound is
achieved if only if the manifold is isometric to an Euclidean ball.
