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

## TL;DR

This paper constructs a minimal surface with positive genus spanning an elongated tetrahedron using triple covers, demonstrating it has less area than the cone and addressing a long-standing open question.

## Contribution

It introduces a novel method employing triple covers to model minimal surfaces with positive genus spanning tetrahedral edges, challenging previous assumptions.

## Key findings

- Existence of a positive genus minimal surface spanning an elongated tetrahedron.
- Constructed surface has strictly less area than the conic surface.
- Method uses BV functions and covering space interpretation.

## 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.