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In this paper, we study $4$-dimensional complete hypersurfaces with$w$-constant mean curvature in the unit sphere. We give a lower bound of thescalar curvature for $4$-dimensional complete hypersurfaces with $w$-constantmean curvature. As a by-product, we give a new proof of the result ofDeng-Gu-Wei under the weaker topological condition.

Complete hypersurfaces with $w$-constant mean curvature in the unit spheres