非等变格罗莫夫环面的存在性
The Existence of Non-Equivariant Gromov Tori
AI总结：
本文针对n≥3时平坦环面嵌入球面的问题，证明存在非等变嵌入格罗莫夫环面，给出该问题的否定答案，推进了最优曲率界相关研究。
AI中文摘要：
本文研究如下问题：若平坦环面$\boldsymbol{\top}^n$等距极小嵌入到球面$\boldsymbol{\text{S}}^N$中，其平移群是否必须延拓为外围球面的等距群？Robert Bryant证明，当$n=2$时答案为肯定；Ying Lu、Peng Wang和Zhenxiao Xie近期证明，当$n\boldsymbol{\text{≥}}3$时，浸入情形下答案为否定，但嵌入情形的该问题仍未解决。此问题与Mikhail Gromov和Anton Petrunin关于最优曲率界的工作密切相关，Petrunin证明，任意环面到单位球的浸入，其法曲率最大值至少为$\boldsymbol{\text{√(3n/(n+2))}}$，Gromov构造的环面族达到该界，这类达到最优界的环面称为“格罗莫夫环面”。本研究首先证明任意格罗莫夫环面本质平坦、位于球面内且在球面内极小，随后确定定义这类环面的充要条件，最终得到核心结果：当$n\boldsymbol{\text{≥}}3$时，存在非等变嵌入格罗莫夫环面，为上述问题提供明确否定答案。
英文摘要：
In this paper, we address the following question: if a flat torus $\mathbb{T}^n$ is isometrically and minimally embedded into a sphere $\mathbb{S}^N$, must its translation group extend to the isometry group of the ambient sphere? As shown by Robert Bryant, for $n=2$ the answer is positive. Furthermore, while Ying Lu, Peng Wang, and Zhenxiao Xie recently demonstrated that the answer is negative for immersions when $n \geq 3$, the question for embeddings remained open. This problem is deeply tied to the work of Mikhail Gromov and Anton Petrunin concerning optimal curvature bounds. Petrunin proved that any immersion of a torus into a unit ball must have a maximum normal curvature of at least $\sqrt{\frac{3n}{n+2}}$. This bound is attained, for example, by families of tori constructed by Gromov. We call the tori that attain this optimal bound "Gromov tori". In this work, we first demonstrate that any Gromov torus is intrinsically flat, lies within a sphere, and is minimal inside it. We then establish the necessary and sufficient conditions for defining these tori. Finally, we present our main result: for dimensions $n \ge 3$, there exists a non-equivariant embedded Gromov torus, which provides a definitive negative answer to the question above.
