嵌入在圆4-球面中的不可定向极小曲面
Nonorientable Minimal Surfaces Embedded in the Round $4$-sphere
AI总结：
该研究证明所有闭不可定向曲面均可极小嵌入圆4-球面，得到面积低于8π的相关曲面，还识别出高度对称族并发现无连续对称群的新单周期极小曲面，发展了等变特征值优化方法。
AI中文摘要：
我们证明了每一个闭的不可定向曲面都可以极小嵌入到$\mathbb{S}^4$中，这尤其提供了$\mathbb{S}^4$中首个已知的带负欧拉示性数的嵌入不可定向极小曲面的例子。我们得到的所有曲面的面积都低于$8\pi$，这可应用于$\mathbb{R}^n$（$n\geq4$）中具有指定拓扑的不可定向曲面极小化Willmore泛函的存在性问题。此外，我们证明了每个不可定向亏格的几何上不同的嵌入数量至少随亏格指数增长。在这些曲面中，我们识别出一个特殊的高度对称族，该族在大亏格极限下收敛到四个半球的并，它们沿一个大圆相交，其极点是正四面体的顶点。该族的重缩放极限给出了$\mathbb{R}^4$中一个新的单周期不可定向极小曲面，这似乎是$\mathbb{R}^4$中首个无连续对称群的完备嵌入不可定向极小曲面的例子。证明进一步发展了我们前期工作中的等变特征值优化方法，该方法应用于作用在不可定向曲面上的精心选取的对称群族。有趣的是，我们还发现了一些自然的曲面与群作用对，对于这些对，不存在使拉普拉斯算子的第一归一化特征值最大化的度量。
英文摘要：
We show that every closed, nonorientable surface can be minimally embedded in $\mathbb{S}^4$, providing in particular the first known examples of embedded, nonorientable minimal surfaces in $\mathbb{S}^4$ with negative Euler characteristic. All of the surfaces we obtain have area below $8π$, which has applications to the existence of nonorientable surfaces minimizing the Willmore functional with prescribed topology in $\mathbb{R}^n$ for $n\geq 4$. Moreover, the number of geometrically distinct embeddings of each nonorientable genus is shown to grow at least exponentially with respect to the genus. Among these surfaces, we identify a distinguished, highly symmetric family which converges in the large-genus limit to a union of four half-spheres, meeting along a great circle, whose poles are vertices of a regular tetrahedron. Rescalings of this family converge to a new singly-periodic nonorientable minimal surface in $\mathbb{R}^4$, which seems to provide the first example of a complete, embedded, nonorientable minimal surface in $\mathbb{R}^4$ without continuous symmetry group. The proofs further develop the equivariant eigenvalue optimization methodology from our earlier work, applied to carefully chosen families of symmetry groups acting on nonorientable surfaces. Interestingly, we also find natural pairs of surfaces and group actions for which there is no metric maximizing the first normalized eigenvalue of the Laplacian.
