We prove that every plane passing through the origin divides an embedded compact free boundary minimal surface of the euclidean 3-ball into exactly two connected surfaces. We also show that if a region in the ball has mean convex boundary and contains a nulhomologous diameter, then this region is the intersection of two closed halfballs. Crucial to both results is the regularity at the corners of currents minimizing a partially free boundary problem, which we get by following ideas by Gruter and Simon.

中文翻译：

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球中自由边界最小曲面的两件式属性

我们证明了每一个通过原点的平面都将欧几里得 3 球的一个嵌入的紧凑自由边界最小表面划分为两个连接的表面。我们还表明，如果球中的一个区域具有平均凸边界并包含一个非同源直径，则该区域是两个封闭半球的交集。这两个结果的关键是电流拐角处的规律性使部分自由边界问题最小化，这是我们通过遵循 Gruter 和 Simon 的想法得到的。