稳定余维1积分Varifold的结构理论及其在模p面积极小超曲面中的应用
A Structure Theory for Stable Codimension 1 Integral Varifolds with Applications to Area Minimising Hypersurfaces mod p
AI总结：
本文建立无低密度经典奇点的稳定余维1平稳积分varifold的结构理论，并用于刻画模p面积极小超曲面在平面切锥点附近的唯一性与广义$C^{1,\alpha}$分支结构。
摘要：
对于任意$Q\in\{\frac{3}{2},2,\frac{5}{2},3,\dotsc\}$，我们为类$\mathcal{S}_Q$建立了一套结构理论；该类由不存在密度小于$Q$的经典奇点的稳定余维1平稳积分varifold组成。该理论包含三个主要定理，分别描述varifold$V\in\mathcal{S}_Q$在以下情形中的性质：（i）$V$接近一个重数为$Q$的平坦圆盘（当$Q$为整数时）；（ii）$V$接近一个整数重数小于$Q$的平坦圆盘；以及（iii）$V$接近一个平稳锥，该锥顶点密度为$Q$，支集为沿一条公共轴相交的3个或更多个半超平面的并集。主要的新结果涉及情形（i），尤其给出了$V\in\mathcal{S}_Q$在密度为$Q$的分支点附近的描述。关于（ii）和（iii）的结果直接来自工作[Wic14]的相应部分（并在第2部分中重现）。 取$Q=p/2$时，这三个定理可方便地应用于任意整数$p\geq2$的余维1可求长模p面积极小流，从而仅凭少量易于检验的信息即可确立这种流$T$的局部结构性质。具体而言，应用情形（i）可知，对于偶数$p$，如果$T$在内点$y$处有一个切锥等于重数为$p/2$的（定向）超平面$P$，那么$P$就是$y$处唯一的切锥，并且$T$在$y$附近由一个$\frac{p}{2}$值函数的图像给出，该函数在某种广义意义下具有$C^{1,\alpha}$正则性。这解决了研究此类流在具有平面切锥的点附近的局部结构时一个基本的遗留开放问题，并推广了自20世纪70年代以来分别由De Giorgi–Allard正则性理论（[All72]）和White结构理论（[Whi79]）已知的该结果在$p=2$和$p=4$情形下的结论。
英文摘要：
For any $Q\in\{\frac{3}{2},2,\frac{5}{2},3,\dotsc\}$, we establish a structure theory for the class $\mathcal{S}_Q$ of stable codimension 1 stationary integral varifolds admitting no classical singularities of density $<Q$. This theory comprises three main theorems which describe the nature of a varifold $V\in \mathcal{S}_Q$ when: (i) $V$ is close to a flat disk of multiplicity $Q$ (for integer $Q$); (ii) $V$ is close to a flat disk of integer multiplicity $<Q$; and (iii) $V$ is close to a stationary cone with vertex density $Q$ and support the union of 3 or more half-hyperplanes meeting along a common axis. The main new result concerns (i) and gives in particular a description of $V\in \mathcal{S}_Q$ near branch points of density $Q$. Results concerning (ii) and (iii) directly follow from parts of the work [Wic14] (and are reproduced in Part 2). These three theorems, taken with $Q=p/2$, are readily applicable to codimension 1 rectifiable area minimising currents mod $p$ for any integer $p\geq 2$, establishing local structure properties of such a current $T$ as consequences of little, readily checked, information. Specifically, applying case (i) it follows that, for even $p$, if $T$ has one tangent cone at an interior point $y$ equal to an (oriented) hyperplane $P$ of multiplicity $p/2$, then $P$ is the unique tangent cone at $y$, and $T$ near $y$ is given by the graph of a $\frac{p}{2}$-valued function with $C^{1,α}$ regularity in a certain generalised sense. This settles a basic remaining open question in the study of the local structure of such currents near points with planar tangent cones, extending the cases $p=2$ and $p=4$ of the result which have been known since the 1970's from the De Giorgi--Allard regularity theory ([All72]) and the structure theory of White ([Whi79]) respectively.
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