A linear isoperimetric inequality for the punctured Euclidean plane

Gromov hyperbolic spaces and the sharp isoperimetric constant

In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is Gromov hyperbolic. As a particular consequence we obtain that Euclidean space is a borderline case for Gromov hyperbolicity in terms of the isoperimetric function. We prove similar results for the linear filling radius inequality. Our results strengthen and generalize theorems of Gromov, Papasoglu and others.

非正曲率中的极小四面体与等周间隙定理

Minimal tetrahedra and an isoperimetric gap theorem in non-positive curvature

本文在CAT(0)空间中研究Lipschitz 2-球面的等周不等式，首次在高维情形证明当欧几里得不等式常数低于尖锐阈值$c_3=1/(6\sqrtπ)$时，空间允许指数任意接近1的不等式，并引入满足线性不等式的极小四面体作为新工具。

阿达马空间中等于及高于渐近秩的线性等周填充不等式

Linear isoperimetric filling inequalities in Hadamard spaces at and above the asymptotic rank

该研究针对具有有限渐近纳塔维数和有限渐近秩的阿达马空间，证明了维度不低于渐近秩的整周满足线性等周填充不等式，将相关结果的指数优化至最优线性情况并推广至任意有限渐近秩。

Spaces with almost Euclidean Dehn function

We prove that any proper, geodesic metric space whose Dehn function grows asymptotically like the Euclidean one has asymptotic cones which are non-positively curved in the sense of Alexandrov, thus are ${\rm CAT}(0)$. This is new already in the setting of Riemannian manifolds and establishes in particular the borderline case of a result about the sharp isoperimetric constant which implies Gromov hyperbolicity. Our result moreover provides a large scale analog of a recent result of Lytchak and the author which characterizes proper ${\rm CAT}(0)$ in terms of the growth of the Dehn function at al…