Triple covers and a non-simply connected surface spanning an elongated tetrahedron and beating the cone
Giovanni Bellettini, Maurizio Paolini, Franco Pasquarelli · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2018
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.