The classical H surfaces of H. A. Schwarz form a 1-parameter family of triply periodic minimal surfaces (TPMS) that are usually described as close relatives to his more famous P surface. However, a crucial distinction between these surfaces is that the P surface belongs to a 5-dimensional smooth family of embedded TPMS of genus three discovered by W. Meeks, while the H surfaces are among the few known examples outside this family. We construct a 2-parameter family of embedded TPMS of genus three that contains the H family and meets the Meeks family. In particular, we prove that H surfaces can be deformed continuously within the space of TPMS of genus three into Meeks surfaces.

中文翻译：

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Schwarz'H 曲面的正交变形族

HA Schwarz 的经典 H 表面形成了三重周期最小表面 (TPMS) 的 1 参数族，通常被描述为与他更为著名的 P 表面的近亲。然而，这些表面之间的一个关键区别是 P 表面属于 W. Meeks 发现的第三属嵌入式 TPMS 的 5 维光滑家族，而 H 表面是该家族之外的少数已知例子之一。我们构建了一个包含 H 家族并满足 Meeks 家族的第三属嵌入式 TPMS 的 2 参数家族。特别地，我们证明了 H 表面可以在属 3 的 TPMS 空间内连续变形为 Meeks 表面。