Abstract
In this note, we use Simons' identity for the Laplacian of the second fundamental form for minimal hypersurfaces to obtain a number of new estimates for the curvatures of stable minimal hypersurfaces M which are immersed in a Riemannian manifold N. Under suitable assumptions on N, Schoen, Simon and Yau found a pointwise bound for the principal cur- vatures of M, provided dim(M) 6 5. In this case, this pointwise bound implies Bernstein's theorem for n 6 5. Schoen, Simon and Yau also gave a simpli ed proof of Simons' result: no non-trivial 6-dimensional stable minimal cones in R7 implies Bernstein's theorem holds for n = 6.