$\R^n$中二维几乎极小集的$C^{1+α}$ -正则性
$C^{1+α}$-Regularity for Two-Dimensional Almost-Minimal Sets in $\R^n$
摘要：
我们给出了一个新的证明和Jean Taylor的结果[Ta]的部分推广，即$\R^3$中2维的Almgren几乎极小集是局部$C^{1+α}$ -等价于极小锥的.证明是相当初级的，但使用了在[D3]中证明的局部分离结果和Reifenberg参数化定理[DDT]的扩展。其核心思想仍然是：如果$X$是单位球面上小Lipschitz图的弧上的锥，而$X$不包含在圆盘中，则可以利用调和函数的图形使$X$变形，从而使其面积大大减小.局部分离的结果被用来减少Lipschitz图的弧上的锥的并集。一个很好的部分证明扩展到最小集的尺寸为2 $\R^n$，但在这种情况下，我们最终的正则性结果$E$可能取决于列表的最小锥作为爆破限制$E$在一个点。
英文摘要：
We give a new proof and a partial generalization of Jean Taylor's result [Ta] that says that Almgren almost-minimal sets of dimension 2 in $\R^3$ are locally $C^{1+α}$-equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in [D3] and an extension of Reifenberg's parameterization theorem [DDT]. The key idea is still that if $X$ is the cone over an arc of small Lipschitz graph in the unit sphere, but $X$ is not contained in a disk, we can use the graph of a harmonic function to deform $X$ and diminish substantially its area. The local separation result is used to reduce to unions of cones over arcs of Lipschitz graphs. A good part of the proof extends to minimal sets of dimension 2 in $\R^n$, but in this setting our final regularity result on $E$ may depend on the list of minimal cones obtained as blow-up limits of $E$ at a point.
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