AI 中文总结
该论文为研究n维面积最小化可求长流T的局部结构，分析平面频率≠2的分支点，得到高阶渐近性、分支集分解及拓扑控制结果，避免使用中心流形，基于平面频率单调性公式论证，为后续研究奠定基础。
AI 中文摘要
这是一系列研究余维数≥2的n维面积最小化可求长流T局部结构新框架的论文的第三部分。第一和第二部分为T引入了固有频率函数——平面频率，并利用其单调性等性质，证明了${\mathcal H}^{n - 2}$-几乎处处的分支点是平面频率至少为$1+\alpha$的快速衰减分支点。本文分析平面频率≠2的分支点，得到：(1)高阶渐近性：在${\mathcal H}^{n - 2}$-几乎处处这样的点，流有大于1的有限阶展开，且余项有精确衰减估计；(2)分支集分解：此类分支点集局部分解为有限多个两两不相交、局部n - 2可求长集（局部有限测度）；(3)拓扑控制：在满足特定平面频率准则的任何分支点附近，T的支撑同胚于n维圆盘且有$C^{1,\mu}$参数化。这里的工作（以及第一和第二部分）避免使用中心流形，而是基于平面频率单调性公式的固有几何论证。第四部分利用中心流形分析平面频率为2的点，此时中心流形变得必要且几何上规范，有额外简化性质。
英文摘要
This is the third part in a series of papers developing a new framework to study the local structure of $n$-dimensional area-minimizing rectifiable currents $T$ of codimension $\geq 2$. Parts I and II introduced an intrinsic frequency function for $T$ -- planar frequency -- and used its monotonicity properties, among other things, to establish that ${\mathcal H}^{n-2}$-a.e. branch point is a rapid-decay branch point where the planar frequency is at least $1 + α$. This paper analyses branch points of planar frequency $\neq 2$. It establishes: (1) higher order asymptotics: at ${\mathcal H}^{n-2}$-a.e. such point, the current admits an expansion of finite order $>1$, with precise decay estimates for the remainder term; (2) branch set decomposition: the set of such branch points locally decomposes into finitely many pairwise disjoint, locally $n-2$ rectifiable sets (of locally finite measure); (3) topological control: near any branch point satisfying a specific planar-frequency criterion, the support of $T$ is homeomorphic to an $n$-dimensional disk and admits a $C^{1, μ}$ parametrization. (Classical complex algebraic examples show that when this frequency criterion fails, the current need not be locally homeomorphic to an $n$-disk). The work here (as well as in parts I & II) avoids the use of center manifolds -- a technically demanding foundational component of the classical Almgren framework -- and uses instead intrinsic geometric arguments based on the monotonicity formula for planar frequency. In part IV, a center manifold is utilised to analyse planar frequency 2 points, where the center manifold becomes necessary and geometrically canonical, satisfying additional simplifying properties. Reduced reliance on center manifolds in our framekwork is necessitated by the structural results it establishes for $T$.
Comments156 pages
