On the fluid ball conjecture

The central set and its application to the Kneser-Poulsen conjecture

The Kneser-Poulsen conjecture says that if a finite collection of balls in a Euclidean (spherical or hyperbolic) space is rearranged so that the distance between each pair of centers does not increase, then the volume of the union of these balls does not increase as well. We give new results about central sets of subsets of a Riemannian manifold and apply these results to prove new special cases of the Kneser-Poulsen conjecture in the two-dimensional sphere and the hyperbolic plane.

由小测地球体积确定的4维空间形式

4-dimensional Space forms as determined by the volumes of small geodesic balls

本文基于张量微积分与经典定理而非拓扑刻画，证明了小测地球体积能确定4维流形是否为空间形式，扩展了Gray-Vanhecke猜想的相关结论。

球体上非正曲率度量的小扰动的局部等周不等式

A local isoperimetric inequality for balls with nonpositive curvature

本文研究了球体在非正曲率度量下的局部等周不等式，证明小扰动不影响等周比，并揭示了保持等周比的条件是相似变换。

Quantitative Volume Space From Rigidity with lower Ricci curvature bound II

This is the second paper of two in a series under the same title ([CRX]); both study the quantitative volume space form rigidity conjecture: a closed $n$-manifold of Ricci curvature at least $(n-1)H$, $H=\pm 1$ or $0$ is diffeomorphic to a $H$-space form if for every ball of definite size on $M$, the lifting ball on the Riemannian universal covering space of the ball achieves an almost maximal volume, provided the diameter of $M$ is bounded for $H\ne 1$. In [CRX], we verified the conjecture for the case that $M$ or its Riemannian universal covering space $\tilde M$ is not collapsed for $H=1$ o…

Approaching the isoperimetric problem in $H^m_{\mathbb{C}}$ via the hyperbolic log-convex density conjecture

We prove that geodesic balls centered at some base point are isoperimetric in the real hyperbolic space $H_{\mathbb R}^n$ endowed with a smooth, radial, strictly log-convex density on the volume and perimeter. This is an analogue of the result by G. R. Chambers for log-convex densities on $\mathbb R^n$. As an application we prove that in any rank one symmetric space of non-compact type, geodesic balls are isoperimetric in a class of sets enjoying a suitable notion of radial symmetry.

Bezdek猜想的一个解

A solution to Bezdek's conjecture

本文证明在n≥2时，给定内径的λ-凸体中λ-凸透镜平均宽度最大，肯定回答Bezdek猜想；附加对称性时解决其任意内蕴体积的类似猜想；还证明固定外接半径的λ-凸体中λ-凸纺锤体平均宽度唯一最小。

Balls Isoperimetric in $\mathbb{R}^n$ with Volume and Perimeter Densities $r^m$ and $r^k$

We have discovered a "little" gap in our proof of the sharp conjecture that in $\mathbb{R}^n$ with volume and perimeter densities $r^m$ and $r^k$, balls about the origin are uniquely isoperimetric if $0 < m \leq k - k/(n+k-1)$, that is, if they are stable (and $m > 0$). The implicit unjustified assumption is that the generating curve is convex.