标题： 关于多重性2平面附近平稳积分变分的性质 显示英文标题

标题： On the Nature of Stationary Integral Varifolds near Multiplicity 2 Planes

摘要： We study stationary integral $n$-varifolds $V$ in the unit ball $B_1(0)\subset\mathbb{R}^{n+k}$. Allard的正则性定理确立了$\epsilon = \epsilon(n,k)\in (0,1)$的存在，使得如果$V$作为变分集与平面$P_0 = \{0\}^k\times\mathbb{R}^n$在接近度上是$\epsilon$，则在$B_{1/2}(0)$中，$V$由一个单个的$C^{1,\alpha}$极小图表示。 然而，当$P_0$以重数$Q\in \{2,3,\dotsc\}$出现时，简单的例子表明，这个结论现在作为一个多值图可能会失败，即使$V$对应于一个面积最小的可测电流。在本工作中，我们研究了这些接近具有重数$Q>1$的平面的$V$的结构，主要关注情况$Q=2$。 We show that an $\epsilon$-regularity theorem holds when $V$ is close, as a varifold, to $P_0$ with multiplicity $2$, provided $V$ satisfies a certain topological structural condition on the part of its support where the density of $V$ is $<2$. 结论是，在 $B_{1/2}(0)$ 中， $V$ 由 $P_0$ 上的 Lipschitz $2$-值函数的图像表示，且 Lipschitz 常数较小；事实上，该函数在一种精确的广义意义上是 $C^{1,\alpha}$，并且满足估计式，这意味着 $B_{1/2}(0)$ 中奇异点的所有切锥都是唯一的，并由 $4$ 半平面的静止并集组成（这些半平面可能形成两个不同平面的并集或一个多重性为 $2$ 的平面）。 定理不需要对具有密度$\geq 2$的$V$做出任何额外假设（这在先验上可能是在$\mathcal{H}^n$-测度下具有高拓扑复杂性的相对较大的集合）。 作为推论，我们证明我们的$\epsilon$-正则性定理无条件适用于任意 Lipschitz 常数的平稳$2$-值 Lipschitz 图，从而得到改进的正则性和统一的先验估计。 显示英文摘要

摘要： We study stationary integral $n$-varifolds $V$ in the unit ball $B_1(0)\subset\mathbb{R}^{n+k}$. Allard's regularity theorem establishes the existence of $\epsilon = \epsilon(n,k)\in (0,1)$ for which if $V$ is $\epsilon$-close (as varifolds) to the plane $P_0 = \{0\}^k\times\mathbb{R}^n$ with multiplicity 1 then, in $B_{1/2}(0)$, $V$ is represented by a single $C^{1,\alpha}$ minimal graph. However, when instead $P_0$ occurs with multiplicity $Q\in \{2,3,\dotsc\}$, simple examples show that this conclusion, now as a multi-valued graph, may fail, even if $V$ corresponds to an area-minimising rectifiable current. In the present work we investigate the structure of such $V$ which are close to planes with multiplicity $Q>1$, focusing primarily on the case $Q=2$. We show that an $\epsilon$-regularity theorem holds when $V$ is close, as a varifold, to $P_0$ with multiplicity $2$, provided $V$ satisfies a certain topological structural condition on the part of its support where the density of $V$ is $<2$. The conclusion then is that, in $B_{1/2}(0)$, $V$ is represented by the graph of a Lipschitz $2$-valued function over $P_0$ with small Lipschitz constant; in fact, the function is $C^{1,\alpha}$ in a precise generalised sense, and satisfies estimates, implying that all tangent cones at singular points in $B_{1/2}(0)$ are unique and comprised of stationary unions of $4$ half-planes (which may form a union of two distinct planes or a single multiplicity $2$ plane). The theorem does not require any additional assumption on the part of $V$ with density $\geq 2$ (which a priori may be a relatively large set in $\mathcal{H}^n$-measure with high topological complexity). As a corollary, we show that our $\epsilon$-regularity theorem applies unconditionally to stationary $2$-valued Lipschitz graphs with arbitrary Lipschitz constant, yielding improved regularity and uniform a priori estimates.