Lf-index bounds for closed f-minimal hypersurfaces☆

Abstract We discuss a two-side closed embedded f -minimal hypersurface Σ in a smooth metric measure space ( M n + 1 , g ¯ , e − f d μ ) and obtain that L f -index of Σ is bounded from below by a constant times the first Betti number with a suitable condition. We also apply the result to f -minimal hypersurfaces in shrinking gradient solitons.