Triple covers and a non-simply connected surface spanning an elongated tetrahedron and beating the cone

Abstract

By using a suitable triple cover we show how to possibly model the construction of a minimal surface with positive genus spanning all six edges of a tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. After a question raised by R. Hardt in the late 1980's, it seems common opinion that an area-minimizing surface of this sort does not exist for a regular tetrahedron, although a proof of this fact is still missing. In this paper we show that there exists a surface of positive genus spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the conic surface.

Covers, soap films and BV functions

“…As discussed in [4] for the example of the tetrahedral wire, also in this example we cannot exclude a priori that a minimizing surface wets the invisible wire: we have already remarked that this is a difficulty present in any example constructed using invisible wires.…”

“…• when S is not smooth, for instance S the one-skeleton of a polyhedron. We refer to [7], [2] and [4] for a more complete description for covers of any (finite) degree.…”

“…Proving that a soap film has no convenience to wet the invisible wires for special choices of their position, seems to be an open problem, not discussed in [7]. We refer to [4] for more. • Instead of describing explicitely the cut and past procedure (as in Section 2) and the parametrizing maps (which becomes more and more complicated as the degree of the cover increases) now it is often convenient to construct the cover first by orienting all portions 12 of the cut, then declaring in a consistent global way the permutations for gluing the sheets along the cut, and finally to use the local triviality of the cover, in order to check the consistency of the gluing.…”

“…The interest in the covering space method is also illustrated in the recent paper [4], where is shown a triple cover of R 3 \ (S ∪ C), S a tetrahedral frame and C two disk boundaries, compatible with a soap film spanning S and having higher topological type, more precisely with two tunnels (see Figure 1 in the case of the regular tetrahedron). fig.…”

“…Date: November 7, 2018. The cover described in [4] has the particular feature of being not normal; in addition, it is constructed using the above mentioned disks. Similar disks were firstly introduced in [7] in other examples, and called invisible wires by the author.…”