AI 中文总结
该研究证实了$\boldsymbol{R}^4$中不存在处处局部度量极小化的光滑严格鞍曲面，构造了支撑在任意小圆盘内的非平凡无穷小等距形变，并给出了一个反例，证明了更高维空间中鞍曲面的度量极小性与三维空间的情况不同。
AI 中文摘要
我们研究欧氏空间中光滑曲面的鞍性质与局部度量极小性之间的关系。Anton Petrunin 和 Stephan Stadler 证明了，$\boldsymbol{R}^3$ 中每一个光滑严格鞍曲面都是局部度量极小化的，并表示期望该现象在 $\boldsymbol{R}^4$ 中不成立。我们证实了这一期望，并证明了一个更强的密度结果：将闭圆盘光滑严格鞍嵌入到 $\boldsymbol{R}^4$ 中的每一个嵌入，都可以在 $C^\text{\textit{\textbf{infty}}}$ 拓扑中被光滑严格鞍嵌入任意逼近，且这些嵌入无处局部度量极小化。特别地，对于所有 $d \boldsymbol{\textbf{≥}} 4$，$\boldsymbol{R}^d$ 中都存在无处局部度量极小化的鞍嵌入。主要方法是构造 $\boldsymbol{R}^4$ 中鞍浸入的非平凡无穷小等距形变，该形变支撑在任意小的圆盘内。我们还通过推导度量极小性的必要条件，给出了一个明确的扭曲二次曲面，它在原点邻域内是鞍曲面，但在该邻域内不满足局部度量极小性。
英文摘要
We study the relation between the saddle property and local metric minimality for smooth surfaces in Euclidean spaces. Anton Petrunin and Stephan Stadler proved that every smooth strictly saddle surface in $\mathbb{R}^3$ is locally metric-minimising and expressed the expectation that this phenomenon fails in $\mathbb{R}^4$. We confirm this expectation and prove a stronger density result: Every smooth strictly saddle embedding of a closed disc into $\mathbb{R}^4$ can be approximated arbitrarily closely in the $C^\infty$-topology by smooth strictly saddle embeddings that are nowhere locally metric-minimising. In particular, nowhere locally metric-minimising saddle embeddings exist in $\mathbb{R}^d$ for every $d \geq 4$. The main ingredient is the construction of non-trivial infinitesimal isometric deformations of saddle immersions in $\mathbb{R}^4$ supported in arbitrarily small discs. We also provide an explicit twisted quadratic surface that is saddle in a neighbourhood of the origin but fails to be locally metric-minimising there, by deriving a necessary condition for metric minimality.
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