小分离器、l^∞-宽度的上界与 systole 几何
Small separators, upper bounds for $l^\infty$-widths, and systolic geometry
AI总结：
该研究探讨闭子流形的欧氏体积与$l^\infty$-宽度对维度的依赖，推导两类$l^\infty$-宽度上界，作为推论得到本质子流形的非可缩闭曲线存在性，还引入了 systole 几何的非线性新方法。
摘要：
我们研究了闭子流形$M^n\subset\mathbb{R}^N$的欧氏体积与其$l^\infty$-宽度$W^{l^\infty}_{n-1}(M^n)$之间不等式对维度的依赖关系，其中$l^\infty$-宽度定义为所有连续映射$\phi:M^n\longrightarrow K^{n-1}\subset\mathbb{R}^N$的$\sup_{x\in M^n}\Vert \phi(x)-x\Vert_{l^\infty}$的下确界。我们证明$W^{l^\infty}_{n-1}(M^n)\leq const\\ \sqrt{n}\\ vol(M^n)^{\frac{1}{n}}$；若余维数$N-n=1$，则$W^{l^\infty}_{n-1}(M^n)\leq \sqrt{3}\\ vol(M^n)^{\frac{1}{n}}$。作为推论，若$M^n\subset\mathbb{R}^N$是本质的（essential），则$M^n$上存在一条非可缩闭曲线，该曲线包含在$\mathbb{R}^N$中边长为$const\\ \sqrt{n}\\ vol^{\frac{1}{n}}(M^n)$且边平行于坐标轴的立方体中；若余维数为1，则立方体边长为$4\\ vol^{\frac{1}{n}}(M^n)$。为证明这些结果，我们引入了一种新的 systole 几何方法，可描述为经典 Federer-Fleming 论证的非线性版本，该方法从$\mathbb{R}^N$中一个专门构造的不与$M^n$相交的非线性$(N-n)$维复形中向外推导。为构造这些复形，我们首先证明了一个运动公式版本，其中对$l^N_\infty$的等距变换取平均（定理3.5），并引入了文献[KORW]和[AK]中近期发现的最优泡沫的高维类比。
英文摘要：
We investigate the dependence on the dimension in the inequalities that relate the Euclidean volume of a closed submanifold $M^n\subset \mathbb{R}^N$ with its $l^\infty$-width $W^{l^\infty}_{n-1}(M^n)$ defined as the infimum over all continuous maps $ϕ:M^n\longrightarrow K^{n-1}\subset\mathbb{R}^N$ of $sup_{x\in M^n}\Vert ϕ(x)-x\Vert_{l^\infty}$. We prove that $W^{l^\infty}_{n-1}(M^n)\leq const\ \sqrt{n}\ vol(M^n)^{\frac{1}{n}}$, and if the codimension $N-n$ is equal to $1$, then $W^{l^\infty}_{n-1}(M^n)\leq \sqrt{3}\ vol(M^n)^{\frac{1}{n}}$. As a corollary, we prove that if $M^n\subset \mathbb{R}^N$ is {\it essential}, then there exists a non-contractible closed curve on $M^n$ contained in a cube in $\mathbb{R}^N$ with side length $const\ \sqrt{n}\ vol^{\frac{1}{n}}(M^n)$ with sides parallel to the coordinate axes. If the codimension is $1$, then the side length of the cube is $4\ vol^{\frac{1}{n}}(M^n)$. To prove these results we introduce a new approach to systolic geometry that can be described as a non-linear version of the classical Federer-Fleming argument, where we push out from a specially constructed non-linear $(N-n)$-dimensional complex in $\mathbb{R}^N$ that does not intersect $M^n$. To construct these complexes we first prove a version of kinematic formula where one averages over isometries of $l^N_\infty$ (Theorem 3.5), and introduce high-codimension analogs of optimal foams recently discovered in [KORW] and [AK].
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