Simons' equation and minimal hypersurfaces in space forms

Let n\geq {}3 be an integer, and let \Sigma ^n be a non-totally geodesic complete minimal hypersurface immersed in the (n+1)-dimensional space form \overline {M}^{n+1}(c), where the constant c denotes the sectional curvature of the space form. If \Sigma ^n satisfies the Simons' equation (3.9), then either (1) \Sigma ^n is a catenoid if c\leq {}0, or (2) \Sigma ^n is a Clifford minimal hypersurface or a compact Ostuki minimal hypersurface if c>0. This paper is motivated by a 2009 work of Tam and Zhou.