Let \(M^3\) be a hypersurface immersed into the 4-dimensional Euclidean sphere \(\mathbb {S}^4\subset \mathbb {R}^5\) and denote by \({\textbf {H}}\) its mean curvature vector field in \(\mathbb {R}^5\) . This paper locally classifies those hypersurfaces that satisfy the condition \(\Box {\textbf {H}}=\lambda {\textbf {H}}\) with \(\lambda \in \mathbb {R}\) , where \(\Box \) denotes the Cheng-Yau operator of the hypersurface. In particular, we prove that (open pieces of) totally geodesic great 3-spheres \(\mathbb {S}^3\subset \mathbb {S}^4\) are the only hypersurfaces in \(\mathbb {S}^4\) which satisfy the condition \(\Box {\textbf {H}}={\textbf {0}}\) . We also obtain that (open pieces of) totally umbilical small 3-spheres \(\mathbb {S}^3(r)\subset \mathbb {S}^4\) , \(0<r<1\) , and the Clifford torus \(\mathbb {S}^2(\sqrt{1/3})\times \mathbb {S}^1(\sqrt{2/3})\subset \mathbb {S}^4\) are the only hypersurfaces in \(\mathbb {S}^4\) which satisfy the condition \(\Box {\textbf {H}}=\lambda {\textbf {H}}\) for a real number \(\lambda \ne 0\) . This characterizes the Clifford torus \(\mathbb {S}^2(\sqrt{1/3})\times \mathbb {S}^1(\sqrt{2/3})\subset \mathbb {S}^4\) as the only non-trivial hypersurface in \(\mathbb {S}^4\) with \(\Box {\textbf {H}}=\lambda {\textbf {H}}\) for a real number \(\lambda \) .
