Chern's Conjecture with Constant Cubic Trace

Abstract: We prove that the values set of (S=|A|^2) attained by closed embedded minimal hypersurfaces in (\mathbb S^{n+1}(1)) with constant (S) and constant (f_3=\operatorname{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.