Integral-Einstein hypersurfaces in spheres

Combining the intrinsic and extrinsic geometry, we generalize Einstein manifolds to Integral-Einstein (IE) submanifolds. A Takahashi-type theorem is established to characterize minimal hypersurfaces with constant scalar curvature (CSC) in unit spheres, which is the main object of the Chern conjecture: such hypersurfaces are isoparametric. For these hypersurfaces, we obtain some integral inequalities with the bounds characterizing exactly the totally geodesic hypersphere, the non-IE minimal Clifford torus $S^{1}(\sqrt{\frac{1}{n}})\times S^{n-1}(\sqrt{\frac{n-1}{n}})$ and the IE minimal CSC hypersurfaces. Moreover, if further the third mean curvature is constant, then it is an IE hypersurface or an isoparametric hypersurface with $g\leq2$ principal curvatures. In particular, all the minimal isoparametric hypersurfaces with $g\geq3$ principal curvatures are IE hypersurfaces. As applications, we also obtain some spherical Bernstein theorems.