The Structure of Singularities in Soap-Bubble-Like and Soap-Film-Like Minimal Surfaces

Abstract

In this paper we provide a complete classification of the local structure of singularities in a wide class of two-dimensional surfaces in R3 collected under the adjective (M, i, a) minimal by Almgren [A3] (see I(8)). The results, Theorems II. 4, IV. 5, IV. 8, are that the singular set of an (M, i, a) minimal set consists of H6lder continuously differentiable curves along which three sheets of the surface meet (Holder continuously) at equal (120?) angles, together with isolated points at which four such curves meet bringing together six sheets of the surface (H6lder continuously) at equal anglesin fact, in a neighborhood of each singular point, the surface is H6lder continuously diffeomorphic to either the surface Y of Figure 1 or the surface T of Figure 2 (both of which are defined in I(11)). The results apply to (idealizations of) many actual surfaces which are governed by surface tension, such as soap films as in Figure 4 and compound soap bubbles as in Figure 3 (and therefore to aggregates of some kinds of biological and metallurgical cells) (Corollary IV. 9 (i), (ii)), and thus are a proof of a result deduced experimentally by Plateau over 100 years ago [P]. They also apply to surfaces which minimize integrals which equal the area integral times some Holder continuous function on R3. A necessary first step in classifying singularities is to determine all possible area minimizing cones (Proposition II. 3). (In 1864 Lamarle claimed to make such a determination but his analysis of the technically most difficult case Figure 12 (p. 503)-was wrong.) Also included is a proof that the surface T of Figure 2 is in fact area minimizing (Theorem IV. 6); it seems to require the full force of Theorem IV.5 and I have never seen it proved elsewhere. The methods of this paper are