  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 are conjectured to be isoparametric in
the Chern conjecture. 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 minimal isoparametric hypersurfaces with $g\geq3$ principal
curvatures are IE hypersurfaces. As applications, we obtain some spherical
Bernstein theorems, including that any embedded closed minimal surface of genus
no more than $\mathfrak{g}$ inside a tubular neighborhood of constant radius
$r(\mathfrak{g})>0$ around an equator in $\mathbb{S}^3$ is an equator.
