Regularity of Minimal Surfaces

Abstract

In this chapter we can finally prove that partial regularity of minimal surfaces; namely we show that the reduced boundary ∂*E is analytic and the only possible singularities must occur in ∂E – ∂*E. Our major task in the following chapters will be to obtain an estimate for the size of ∂E – ∂*E. As mentioned before, the main step in the regularity theory is the De Giorgi lemma. We show that a minimal surface is regular at points which satisfy the hypotheses of the lemma. Obviously by the definition of reduced boundary this must include all the points in ∂*E and we show that the converse also holds and further that ∂*E is relatively open in ∂E.