论文标题
封闭的Riemannian流形的扫荡
Sweepouts of closed Riemannian manifolds
论文作者
论文摘要
我们表明，对于每一个封闭的riemannian歧管，都存在一个连续的$ 1 $ cycles（定义为有限的分离式封闭曲线集合），该家族被一个球参数化并扫除了整个流形，因此所有连接的封闭曲线的长度以使数量（或二战）和$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n时。该结果的另一种形式涉及对格罗莫夫（Gromov）对扫描腰部的定义进行修改，其中参数空间可以是任何有限的多面体（而不一定是伪曼夫）。我们证明，封闭的riemannian歧管的所谓多面体$ 1 $维腰部等于其填充半径，最多最多是一个恒定因素。我们还建立了某些同源性类别的多面体$ 1 $ WAIST的上限，以环境歧管的数量或直径为单位。此外，我们使用同源填充函数对较高维度的Polyhedra进行了对这些结果的概括。最后，我们证明了封闭的riemannian歧管的填充半径和超透明性可以是任意分开的。
We show that for every closed Riemannian manifold there exists a continuous family of $1$-cycles (defined as finite collections of disjoint closed curves) parametrized by a sphere and sweeping out the whole manifold so that the lengths of all connected closed curves are bounded in terms of the volume (or the diameter) and the dimension $n$ of the manifold, when $n \geq 3$. An alternative form of this result involves a modification of Gromov's definition of waist of sweepouts, where the space of parameters can be any finite polyhedron (and not necessarily a pseudomanifold). We demonstrate that the so-defined polyhedral $1$-dimensional waist of a closed Riemannian manifold is equal to its filling radius up to at most a constant factor. We also establish upper bounds for the polyhedral $1$-waist of some homology classes in terms of the volume or the diameter of the ambient manifold. In addition, we provide generalizations of these results for sweepouts by polyhedra of higher dimension using the homological filling functions. Finally, we demonstrate that the filling radius and the hypersphericity of a closed Riemannian manifold can be arbitrarily far apart.
