Some remarks on the geometry of austere manifolds

Abstract

We prove several structure theorems about the special class of minimal submanifolds which Harvey and Lawson have called “austere” and which arose in connection with their foundational work on calibrations. The condition of austerity is a pontwise condition on the second fundamental form and essentially requires that the non-zero eigenvalues of the second fundamental form in any normal direction at any point occur in oppositely signed pairs. We solve the pointwise problem of describing the set of austere second fundamental forms in dimension at most four and the local problem of describing the austere three-folds in Euclidean space in all dimensions.