Smooth compactness of ƒ-minimal hypersurfaces with bounded ƒ-index

Let (Mⁿ⁺¹, g, e−ƒ dμ) be a complete smooth metric measure space with 2 ≤ n ≤ 6 and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded ƒ-minimal hypersurfaces in M with uniform upper bounds on ƒ-index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in Rⁿ⁺¹ with 2 ≤ n ≤ 6. We also prove some estimates on the ƒ-index of ƒ-minimal hypersurfaces.