arXiv 1907.044252019-07math.DG

球上自由边界极小曲面的两段性质

A two-piece property for free boundary minimal surfaces in the ball

Vanderson Lima, Ana Menezes

备注：: Accepted for publication at Transactions of the American Mathematical Society. We added an appendix with an application of Serrin's Maximum Principle, and strengthened the regularity at the corners. We also improved the presentation of the paper

摘要：

证明了欧氏$3$ -球的嵌入紧致自由边界极小曲面的每一个过原点的平面都恰好分成两个连通曲面。我们还证明了，如果球中的一个区域具有平均凸边界，且包含一个零同调直径，则该区域是一个闭半球。此外，我们证明了在角落处的正则性电流极小化的部分自由边界问题的思想由Grüter和Simon。我们的第一个结果证明了弗雷泽和李的一个猜想。

英文摘要：

We prove that every plane passing through the origin divides an embedded compact free boundary minimal surface of the euclidean $3$-ball in exactly two connected surfaces. We also show that if a region in the ball has mean convex boundary and contains a nullhomologous diameter, then this region is a closed halfball. Moreover, we prove the regularity at the corners of currents minimizing a partially free boundary problem by following ideas by Grüter and Simon. Our first result gives evidence to a conjecture by Fraser and Li.

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