双曲性、CAT（-1）-空间与Ptolemy不等式

The Quasi-hyperbolicity Constant of a Metric Space

We introduce the quasi-hyperbolicity constant of a metric space, a rough isometry invariant that measures how a metric space deviates from being Gromov hyperbolic. This number, for unbounded spaces, lies in the closed interval $[1,2]$. The quasi-hyperbolicity constant of an unbounded Gromov hyperbolic space is equal to one. For a CAT$(0)$-space, it is bounded from above by $\sqrt{2}$. The quasi-hyperbolicity constant of a Banach space that is at least two dimensional is bounded from below by $\sqrt{2}$, and for a non-trivial $L_p$-space it is exactly $\max\{2^{1/p},2^{1-1/p}\}$. If $0 < α< 1$ …

CAT(0)立方复形中的循环双曲性

Cyclic hyperbolicity in CAT(0) cube complexes

本文证明余紧特殊群不含F2×F2当且仅当循环双曲，并在CAT(0)立方复形几何作用群中给出结构定理和强Tits替代，推论SQ-普遍性。

Higher rank hyperbolicity

The large-scale geometry of hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. We prove a number of closely analogous results for spaces of rank $n \ge 2$ in an asymptotic sense, under some weak assumptions reminiscent of nonpositive curvature. For this purpose we replace quasi-geodesic lines with quasi-minimizing (locally finite) $n$-cycles of $r^n$ volume growth; prime examples include $n$-cycles associated with $n$-quasif…

定量猜测测地线定理

A Quantitative Guessing Geodesics Theorem

本文提出猜测测地线定理的定量版本，补充双曲性常数显式估计，并将其应用于估计CAT(0)空间的curtain模型的双曲性常数。

CAT(0) 空间与等距同构的 Curtain 模型

Curtain Model for CAT(0) Spaces and Isometries

本文研究了固定 CAT(0) 空间中 curtain 模型的等距同构动力学，完全分类了半单等距同构行为，证明收缩性与 loxodromic 等价，并排除了多数抛物作用以支持 ping-pong 技术的应用。

Curvature of the space of stability conditions

Motivated by the study of the autoequivalence group of triangulated categories via isometric actions on metric spaces, we consider curvature properties (CAT(0), Gromov hyperbolic) of the space of Bridgeland stability conditions with the canonical metric defined by Bridgeland. We then prove that the metric is neither CAT(0) nor hyperbolic, and the quotient metric by the natural $\mathbb{C}$-action is not CAT(0) in case of the Kronecker quiver. Moreover, we also show the hyperbolicity of pseudo-Anosov functors defined by Dimitrov-Haiden-Katzarkov-Kontsevich, which yields the lower-bound of entro…

$\mathbb{C}$-凸域上的Gromov双曲性、有限型和次椭圆性

Gromov hyperbolicity, finite type, and subellipticity on $\mathbb{C}$-convex domains

本文证明有界光滑$\mathbb{C}$-凸域的Kobayashi距离Gromov双曲性、D'Angelo型有限性与$\bar{\partial}$-Neumann问题次椭圆估计三者等价，并给出相关指数与边界等价性结果。

Gromov双曲空间中的重心与大数定律

Barycenters and a law of large numbers in Gromov hyperbolic spaces

本文研究Gromov双曲空间中概率测度的重心，证明收缩性质、有限均匀分布重心的确定性逼近和大数定律，将CAT(0)空间结果推广至含双曲常数附加项的情形。

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…