极小叶、余维一稳定范数与Bangert的一个问题
Minimal foliations, codimension-one stable norms, and a question of Bangert
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中文总结 AI 辅助
该研究计算环面上等距群余维一度量的余维一稳定范数，对Bangert的问题给出否定答案，在三维环面构造无限维非平坦度量族，证明欧氏稳定范数与体积数据在立方平坦度量附近非局部单射。
中文摘要 AI 辅助
我们计算了环面上一类自然的等距群余维一（cohomogeneity-one）度量的余维一稳定范数。在每个维度n≥3，该公式给出光滑非平坦度量，其中每个本原余维一同调类由校准环面的叶状结构表示，从而对Bangert的问题给出否定答案。在三维环面（$\boldsymbol{\text{T}}^3$）上，我们构造了无限维非平坦度量族，其馀维一稳定范数与单位立方平坦环面的完全一致且总体积固定。一个显式双参数子族包含两两非等距的度量，这些例子还表明，欧氏稳定范数与体积数据在立方平坦度量附近不是局部单射的。
英文摘要
We compute the codimension-one stable norm for a natural class of cohomogeneity-one metrics on tori. In every dimension $n\ge3$, the formula yields smooth nonflat metrics for which each primitive codimension-one homology class is represented by a foliation of calibrated tori, giving a negative answer to a question of Bangert. On $\mathbb T^3$, we construct an infinite-dimensional family of nonflat metrics whose codimension-one stable norm agrees exactly with that of the unit cubic flat torus and whose total volume is fixed. An explicit two-parameter subfamily contains pairwise non-isometric metrics. These examples also show that the Euclidean-stable-norm-and-volume data are not locally injective near the cubic flat metric. Conversely, among smooth metrics on $\mathbb T^3$ admitting a free isometric circle action and having the cubic Euclidean codimension-one stable norm, we prove that volume is at most one, with equality only for the cubic flat metric up to an isometry isotopic to the identity.
发表机构
- The University of Chicago（芝加哥大学）
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