Chern's Conjecture with Constant Cubic Trace

We prove that the values set of $S=\mathrm{\mid }A{\mathrm{\mid }}^2$ attained by closed embedded minimal hypersurfaces in ${\mathbb{S}}^{n+1}(1)$ with constant $S$ and constant $f_3=\mathrm{tr}(A^3)$ is locally finite, where $A$ denotes the shape operator. Neither the topology of the hypersurface nor the value of $f_3$ is fixed.