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.

中文翻译：

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三重覆盖和非简单连接的表面跨越一个细长的四面体并击败圆锥体

通过使用合适的三重覆盖，我们展示了如何构建一个具有正属的最小表面，该表面跨越四面体的所有六个边，在 BV 函数空间中工作并将薄膜解释为覆盖中 Caccioppoli 集的边界空间。在 1980 年代后期 R. Hardt 提出一个问题之后，似乎普遍认为正四面体不存在这种面积最小化表面，尽管仍然缺少对这一事实的证明。在本文中，我们证明存在一个正属曲面，它跨越细长四面体的边界，其面积严格小于圆锥曲面的面积。