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

Let (Mⁿ⁺¹, g, e^−f 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 f-minimal hypersurfaces in M with uniform upper bounds on f-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 f-index of f-minimal hypersurfaces.