  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.
