Eigenvalue estimate and compactness for closed $f$-minimal surfaces

Let $\Omega$ be a bounded domain with convex boundary in a complete noncompact Riemannian manifold with Bakry-\'Emery Ricci curvature bounded below by a positive constant. We prove a lower bound of the first eigenvalue of the weighted Laplacian for closed embedded $f$-minimal hypersurfaces contained in $\Omega$. Using this estimate, we prove a compactness theorem for the space of closed embedded $f$-minimal surfaces with the uniform upper bounds of genus and diameter in a complete $3$-manifold with Bakry-\'Emery Ricci curvature bounded below by a positive constant and admitting an exhaustion by bounded domains with convex boundary.