We add two new 1-parameter families to the short list of known embedded triply periodic minimal surfaces of genus 4 in $\mathbb{R}^3$. Both surfaces can be tiled by minimal pentagons with two straight segments and three planar symmetry curves as boundary. In one case (which has the appearance of the CLP surface of Schwarz with an added handle) the two straight segments are parallel, while they are orthogonal in the second case. The second family has as one limit the Costa surface, showing that this limit can occur for triply periodic minimal surfaces. For the existence proof we solve the 1-dimensional period problem through a combination of an asymptotic analysis of the period integrals and geometric methods. v2: corrected embeddedness proof. Minor further improvements.

中文翻译：

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属 4 的两个新嵌入的三重周期性最小曲面

我们将两个新的 1 参数族添加到 $\mathbb{R}^3$ 中第 4 属的已知嵌入三重周期性最小曲面的短列表中。两个表面都可以用最小五边形平铺，其中两个直线段和三个平面对称曲线作为边界。在一种情况下（具有 Schwarz 的 CLP 表面并增加了手柄），两个直线段是平行的，而在第二种情况下它们是正交的。第二个族将 Costa 曲面作为一个极限，这表明该极限可以出现在三重周期性极小曲面上。对于存在性证明，我们通过对周期积分的渐近分析和几何方法相结合来解决一维周期问题。v2：更正的嵌入性证明。小的进一步改进。