Regularity of directed minimal Cones

Here we prove a Regularity Theorem extending the one of the Diploma Thesis to higher dimensions provided the singular set is small. The proof is a geometric one. Furthermore, we introduce the definition of a directed cone which allows to avoid terms as for almost every x, ... . The result corresponds to a result of DeGiorgi (Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa (3) 19 (1963), 79-85) as was pointed out to me later on.

This paper deals with directed minimal cones in Rⁿ⁺¹ which have at most one singularity. We show that such cones are -- besides the trivial cases the empty set and Rⁿ⁺¹ -- half spaces. Using blow-up techniques we show how this result can be used to get C^1,-regularity for the measure-theoretic boundary of almost minimal Caccioppoli sets which are representable as subgraphs in R⁸.