超临界刘维尔量子引力度量的测地线的正则性与汇合性
Regularity and confluence of geodesics for the supercritical Liouville quantum gravity metric
AI总结：
本文证明超临界刘维尔量子引力度量球的互补连通分量边界是Jordan曲线且紧致有限维，并利用此正则性将测地线汇合性结果推广至超临界情形。
摘要：
设 $h$ 为平面高斯自由场，$D_h$ 为与 $h$ 相关的超临界刘维尔量子引力（LQG）度量。此类度量作为超临界刘维尔首达渗流（Ding-Gwynne, 2020）的子序列缩放极限出现，对应于物质中心荷 $\mathbf{c}_{\mathrm M} \in (1,25)$ 的值。我们证明，几乎必然地，$D_h$-度量球的每个互补连通分量的边界是一条 Jordan 曲线，并且关于 $D_h$ 是紧致且有限维的。这与 $D_h$-度量球的整体边界形成对比，后者关于 $D_h$ 是非紧致且无限维的（Pfeffer, 2021）。利用我们对 $D_h$-度量球的互补连通分量边界的正则性结果，我们将 Gwynne-Miller（2019）的测地线汇合性结果推广到超临界刘维尔量子引力的情形。这些结果表明，具有相同起点和不同目标点的两条 $D_h$-测地线在非平凡的初始时间区间内重合。
英文摘要：
Let $h$ be the planar Gaussian free field and let $D_h$ be a supercritical Liouville quantum gravity (LQG) metric associated with $h$. Such metrics arise as subsequential scaling limits of supercritical Liouville first passage percolation (Ding-Gwynne, 2020) and correspond to values of the matter central charge $\mathbf{c}_{\mathrm M} \in (1,25)$. We show that a.s. the boundary of each complementary connected component of a $D_h$-metric ball is a Jordan curve and is compact and finite-dimensional with respect to $D_h$. This is in contrast to the \emph{whole} boundary of the $D_h$-metric ball, which is non-compact and infinite-dimensional with respect to $D_h$ (Pfeffer, 2021). Using our regularity results for boundaries of complementary connected components of $D_h$-metric balls, we extend the confluence of geodesics results of Gwynne-Miller (2019) to the case of supercritical Liouville quantum gravity. These results show that two $D_h$-geodesics with the same starting point and different target points coincide for a non-trivial initial time interval.
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