In 1968, Ringel and Youngs confirmed the last open case of the Heawood Conjecture by determining the genus of every complete graph $K_n$. In this paper, we investigate the minimum genus embeddings of the complete 3-uniform hypergraphs $K_n^3$. Embeddings of a hypergraph H are defined as the embeddings of its associated Levi graph $L_H$ with vertex set $V(H)\bigsqcup E(H)$, in which $v\in V(H)$ and $e\in E(H)$ are adjacent if and only if v and e are incident in H. We determine both the orientable and the non-orientable genus of $K_n^3$ when n is even. Moreover, it is shown that the number of non-isomorphic minimum genus embeddings of $K_n^3$ is at least $2^{\frac{1}{4}{n}^2\log n(1-o(1))}$. The construction in the proof may be of independent interest as a design-type problem.

1968年，Ringel and Youngs通过确定每个完整图的属来确认了Heawood猜想的最后一个未决案例 ${ķ}_{ñ}$。在本文中，我们研究了完整的3一致超图的最小属嵌入${ķ}_{ñ}^3$。超图H的嵌入定义为其相关Levi图的嵌入${\mathrm{大号}}_H$ 带有顶点集 $V（H）\bigsqcup Ë（H）$，其中 $v\in V（H）$ 和 $Ë\in Ë（H）$当且仅当v和e入射在H中时，相邻。我们确定的可定向和不可定向属${ķ}_{ñ}^3$当n为偶数时 而且，证明了的非同构最小属嵌入数${ķ}_{ñ}^3$ 至少是 $2^{\frac{\mathrm{1个}}{4}{ñ}^2\mathrm{日志}ñ（\mathrm{1个}-Ø（\mathrm{1个}））}$。作为设计类型的问题，证明中的构造可能会引起人们的关注。