拟双曲域是CAT(2)的
Quasihyperbolic domains are CAT(2)
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中文总结 AI 辅助
本文证明欧几里得空间真子域在拟双曲度量下是CAT(2)的，从而在所有维度上解决了Väisälä关于拟双曲测地线的多个猜想，并回答了Gehring-Vuorinen及Martio-Väisälä关于球面正则性和凸性的问题。
中文摘要 AI 辅助
我们证明了欧几里得空间中配备拟双曲度量的每个真子域都是CAT(2)的。这一CAT性质正面解决了Väisälä关于拟双曲测地线的唯一性、延拓、拟双曲凸性和局部测地线猜想，在所有维度上均成立，将Herron的二维结果推广到了任意维度。截面曲率界是最优的，这由在至少三维空间中两个点的补集所证明。我们还证明了在任何拟双曲域中，半径不超过$\pi/\sqrt{2}$的拟双曲球面与欧几里得球面是$\mathcal{C}^{1,\frac{1}{2}}$微分同胚的。因此，我们回答了Gehring和Vuorinen关于此类球面正则性的一个问题。类似的技术正面解决了Väisälä关于半径不超过$\arctan(\sqrt{2})/\sqrt{2}$的拟双曲球的欧几里得凸性的猜想。我们确立了在所有维度上，凸域上的拟双曲度量是CAT(0)的。这正面回答了Martio和Väisälä关于此类域上拟双曲球的拟双曲凸性的一个问题。最后，我们证明了拟双曲域的变量Alexandrov曲率下界可以自改进为全局下界$-1$，并且等价于该域的凹性。
英文摘要
We prove that every proper subdomain of the Euclidean space equipped with its quasihyperbolic metric is CAT(2). The CAT property leads to a positive resolution of Väisälä's uniqueness, prolongation, quasihyperbolic convexity, and local geodesic conjectures on quasihyperbolic geodesics affirmatively on all dimensions, extending the two-dimensional results by Herron. The sectional curvature bound is optimal as demonstrated by the complement of two points in dimensions at least three. We also show that in any quasihyperbolic domain, quasihyperbolic spheres up to radius $π/\sqrt{2}$ are $\mathcal{C}^{1,\frac{1}{2}}$-diffeomorphic to the Euclidean sphere. Consequently, we answer a question by Gehring and Vuorinen on the regularity of such spheres. Similar techniques lead to a positive resolution of Väisälä's conjecture on the Euclidean convexity of quasihyperbolic balls up to radius $\arctan(\sqrt{2})/\sqrt{2}$. We establish that in all dimensions, the quasihyperbolic metric is CAT(0) on convex domains. This leads to a positive answer to a question by Martio and Väisälä on the quasihyperbolic convexity of quasihyperbolic balls on such domains. Finally, we prove that a variable Alexandrov curvature lower bound for a quasihyperbolic domain self-improves to a global lower bound of $-1$ and is equivalent to the concavity of the domain.
发表机构
- University of Helsinki（赫尔辛基大学）
- University of Jyväskylä（于韦斯屈莱大学）
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