Eigenvalue estimate and compactness for closedf-minimal surfaces

Abstract

Let 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 em- bedded f-minimal hypersurfaces contained in . 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.