Urysohn宽度与手术

Urysohn宽度与宏观标量曲率

Urysohn width and macroscopic scalar curvature

本文研究了四维及以上流形中Urysohn宽度与宏观标量曲率的关系，通过新估计和规则概念证明了Gromov猜想的宏观版本不成立。

Local-to-global Urysohn width estimates

The notion of the Urysohn $d$-width measures to what extent a metric space can be approximated by a $d$-dimensional simplicial complex. We investigate how local Urysohn width bounds on a riemannian manifold affect its global width. We bound the $1$-width of a Riemannian manifold in terms of its first homology and the supremal width of its unit balls. Answering a question of Larry Guth, we give examples of $n$-manifolds of considerable $(n-1)$-width in which all unit balls have arbitrarily small $1$-width. We also give examples of topologically simple manifolds that are locally nearly low-dimen…

Waist of maps measured via Urysohn width

We discuss various questions of the following kind: for a continuous map $X \to Y$ from a compact metric space to a simplicial complex, can one guarantee the existence of a fiber large in the sense of Urysohn width? The $d$-width measures how well a space can be approximated by a $d$-dimensional complex. The results of this paper include the following. 1) Any piecewise linear map $f: [0,1]^{m+2} \to Y^m$ from the unit euclidean $(m+2)$-cube to an $m$-polyhedron must have a fiber of $1$-width at least $\frac{1}{2βm +m^2 + m + 1}$, where $β= \sup_y \text{ rk } H_1(f^{-1}(y))$ measures the topolo…

正各向同性曲率下的刚性与柔性

Rigidity and flexibility under positive isotropic curvature

本文构造了具有任意大Urysohn宽度和稳定极小圆盘半径的4-PIC度量，否证了正各向同性曲率下的宽度与圆盘半径界，并在闭偶维流形上证明了尖锐的谱估计及等号刚性。