Eigenvalue estimate and compactness for closedf-minimal surfaces

Let be a bounded domain with convex boundary in a complete noncompact Riemannian manifold with Bakry-Émery Ricci curvature bounded below by a positive constant.We prove a lower bound on the first eigenvalue of the weighted Laplacian for closed embedded f -minimal hypersurfaces contained in .Using this estimate, we prove a compactness theorem for the space of closed embedded f -minimal surfaces with uniform upper bounds on genus and diameter in a complete 3-manifold with Bakry-Émery Ricci curvature bounded below by a positive constant and admitting an exhaustion by bounded domains with convex boundary.where x is the position vector in ‫ޒ‬ nC1 .Hence a self-shrinker is an f -minimal hypersurface † with f D jxj 2 =4 (see more information on self-shrinkers in [Colding and Minicozzi 2012a] and references therein).In the study of f -minimal hypersurfaces, it is convenient to consider the ambient space as a smooth metric measure space .M; N g; e f d /, where d is the volume