A curvature characterization of the Cartan minimal hypersurface in ${\mathbb{S}}^5$

Lawson showed that a non-totally geodesic Einstein minimal hypersurface in ${\mathbb{S}}^5$ is congruent to the Clifford hypersurface ${\mathbb{S}}^2(1\mathrm{/}\sqrt{2})\times {\mathbb{S}}^2(1\mathrm{/}\sqrt{2})\mathrm{.}$ It is also known, by work of Cartan and Ôtsuki, that a non-totally geodesic locally conformally flat minimal hypersurface in ${\mathbb{S}}^5$ is of Ôtsuki type, including the Clifford hypersurface ${\mathbb{S}}^1(1\mathrm{/}2)\times {\mathbb{S}}^3(\sqrt{3}\mathrm{/}2)\mathrm{.}$ In this paper we study closed minimal hypersurfaces $M$ in ${\mathbb{S}}^5$ satisfying $\mathrm{\mid }W{\mathrm{\mid }}^2=2\mathrm{\mid }\stackrel{}{\mathrm{Ric}}{\mathrm{\mid }}^2,$ where $W$ is the Weyl tensor and $\stackrel{}{\mathrm{Ric}}$ is the trace-free Ricci tensor. We call this the Euler-balanced condition. We prove that such a hypersurface is either totally geodesic or congruent to the Cartan minimal hypersurface.