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In this paper, we prove that a noncompact complete hypersurface with finiteweighted volume, weighted mean curvature vector bounded in norm, andisometrically immersed in a complete weighted manifold is proper. In addition,we obtain an estimate for $f$-stability index of a constant weighted meancurvature hypersurface with finite weighted volume and isometrically immersedin a shrinking gradient Ricci soliton that admits at least one parallel fieldglobally defined. For such hypersurface, we still give a necessary conditionfor equality to be achieved in the estimate obtained.

The $f$-Stability Index of the Constant Weighted Mean Curvature Hypersurfaces in Gradient Ricci Solitons