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

Abstract

Lawson showed that a non-totally geodesic Einstein minimal hypersurface in $\mathbb S^5$ is congruent to the Clifford hypersurface $\mathbb S^2(1/\sqrt2)\times \mathbb S^2(1/\sqrt2).$ It is also known, by work of Cartan and \^{O}tsuki, that a non-totally geodesic locally conformally flat minimal hypersurface in $\mathbb S^5$ is of \^{O}tsuki type, including the Clifford hypersurface $\mathbb S^1(1/2)\times \mathbb S^3(\sqrt3/2).$ In this paper we study closed minimal hypersurfaces $M$ in $\mathbb S^5$ satisfying $|W|^2=2|\mathring{\operatorname{Ric}}|^2,$ where $W$ is the Weyl tensor and $\mathring{\operatorname{Ric}}$ 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.