Curvature estimates for minimal hypersurfaces

Abstract

In [12] J. Simons initiated a study of minimal cones from a more differential geometric point of view than had previously been attempted.One of Simons' main results was an identity for the Laplacian of the second fundamental form of minimal hyper-surfaces.Coupling this identity with an analysis of the first eigenvalue of a certain differential operator, he was able to prove that no non-trivial n-dimensional stable minimal cones exist in R n+l for n <6.He was thus able to demonstrate that any boundary of least area in R n~+l, n ~<6, must in fact be a hyperplane, because Fleming [7] had demonstrated that the non-existence of non-trivial stable minimal cones in R ~ implies the result that the only boundaries of least area in R n are the hyperplanes.Simons was in fact able to deduce that, for n ~< 7, the only entire solutions of the minimal surface equation n 2 n -an -~u au ~u