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.

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