奇数分支阻碍重数为一的积分流结构

中文摘要 AI 辅助

Giuliano Basso 曾问：每个 $n$ 维积分流空间是否都允许一个积分流结构，其具有相同的特征集且几乎处处重数为一？我们在维度一就给出了否定答案。障碍是奇数价分支点处的奇偶性现象。更精确地说，如果一个一维积分流在孤立顶点 $v$ 的相邻臂上具有单位重数，那么它在 $v$ 处的边界系数与 $°(v)$ 具有相同的奇偶性。因此，无限多个奇数价顶点迫使每个全支撑单位重数流的边界质量无限。我们构造了一个紧致的、测地线、加倍、$1$-Ahlfors 正则的图状连续统 $G$，其长度有限，并且存在一个无边界流 $T\in\I_1(G)$ 使得 $\set(T)=G$，而任何 $S\in\I_1(G)$ 且 $\set(S)=G$ 都不能在 $\Hh^1$-几乎处处具有重数一。我们对 $G$ 上的所有全支撑积分环进行了分类。特别地，最小可能的本质最大重数恰好是二，且此类环的最小质量为 $3/2$。对于有限度量图，我们确定了一个互补的尖锐缺陷：单位重数流的最小边界质量等于奇数度顶点的数量。最后，通过与平坦环面取乘积并利用精确的 Ambrosio--Kirchheim 切片表示，我们在每个维度都获得了无边界紧致测地线 Ahlfors-正则的反例。

英文摘要

Giuliano Basso asked whether every $n$-dimensional integral current space admits an integral current structure with the same characteristic set and multiplicity one almost everywhere. We give a negative answer already in dimension one. The obstruction is a parity phenomenon at odd-valence branch points. More precisely, if a one-dimensional integral current has unit multiplicity on the arms incident to an isolated vertex $v$, then the coefficient of its boundary at $v$ has the same parity as $°(v)$. Hence infinitely many odd-valence vertices force infinite boundary mass for every full-support unit-multiplicity current. We construct a compact geodesic, doubling, $1$-Ahlfors regular graph-like continuum $G$ of finite length and a boundaryless current $T\in\I_1(G)$ with $\set(T)=G$, while no $S\in\I_1(G)$ with $\set(S)=G$ can have multiplicity one $\Hh^1$-almost everywhere. We classify all full-support integral cycles on $G$. In particular, the least possible essential maximal multiplicity is exactly two, and the minimum mass of such a cycle is $3/2$. For finite metric graphs we identify a complementary sharp defect: the minimum boundary mass among unit-multiplicity currents equals the number of odd-degree vertices. Finally, by taking products with flat tori and using the precise Ambrosio--Kirchheim slice representation, we obtain boundaryless compact geodesic Ahlfors-regular counterexamples in every dimension.

发表机构

• Institute of Applied Mathematics, Shenzhen Polytechnic University（深圳职业技术大学应用数学研究所）

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