Triple coverings and a genus-two surface spanning an elongated tetrahedron and beating the cone
Giovanni Bellettini, Maurizio Paolini, Franco Pasquarelli · arXiv (Cornell University) · 2017
Using a suitable triple covering space it is possible to model the construction of a minimal surface with genus two 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. The possibility of using covering spaces for minimal surfaces was first proposed by Brakke. After a question raised by Bob 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 genus two spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the conic surface.