On mean curvature integrals of the outer parallel convex body of constant width

等宽物体的典型曲率特性

Typical curvature behaviour of bodies of constant width

已知在Baire范畴意义下典型的$n$维凸体表现出简单但高度非直观的曲率行为：在几乎所有的边界点处，在测度意义下，所有曲率为零，但也存在一个稠密且不可数的边界点集合，在该集合处所有曲率为无穷大。本文的目的是找到一个对应的典型凸体，给定的恒定宽度的现象。这样的物体不可能有零曲率。一个主要结果是，对于一个典型的等宽$1$的$n$维凸体（不失一般性），在几乎所有的边界点上，在测度意义下，所有的曲率都等于$1$ . (In相反，注意，宽度为$1$的球具有半径$1/2$，因此其所有曲率都等于$2$。）由于常数宽度的性质是线性的闵可夫斯基加法，证明需要求助于线性曲率的概念，这是由切线曲率半径提供的。

Convex bodies of constant width in spaces of constant curvature and the extremal area of Reuleaux triangles

Extending Blaschke and Lebesgue's classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width $D$. We prove an essentially optimal stability version of this statement in each of the three types of surfaces of constant curvature. In addition, we summarize the fundamental properties of convex bodies of constant width in spaces of constant curvature, and provide a characterization in the hyperbolic case in terms of horospheres.

关于具有正数量曲率的完备3-流形中的腰围与宽度不等式

On the waist and width inequality in complete 3-manifolds with positive scalar curvature

本文证明了数量曲率有正下界的完备非紧3-流形上存在由面积和直径受控的曲面构成的奇异叶状结构，从而给出腰围与宽度不等式。

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。

Uryson width of three dimensional mean convex domain with non-negative Ricci curvature

We prove that for every three dimensional manifold with nonnegative Ricci curvature and strictly mean convex boundary, there exists a Morse function so that each connected component of its level sets has a uniform diameter bound, which depends only on the lower bound of mean curvature. This gives an upper bound of Uryson 1-width for those three manifolds with boundary.

宏观数量曲率与余维2宽度

Macroscopic scalar curvature and codimension 2 width

本文证明，在特定宏观曲率条件下（球体积和同调约束），具有有限生成第一同调群的完备三维黎曼流形的宏观维数为1。