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.
Cite this article
Giovanni Bellettini, Maurizio Paolini, Franco Pasquarelli, Triple covers and a non-simply connected surface spanning an elongated tetrahedron and beating the cone. Interfaces Free Bound. 20 (2018), no. 3, pp. 407–436DOI 10.4171/IFB/407