论文标题
liouville量子重力中的度量球体积
Volume of metric balls in Liouville quantum gravity
论文作者
论文摘要
我们研究了liouville量子重力（LQG）中度量球的体积。自$γ\ in（0,2）$中，自Kahane（1985）和Molchan（1996）的早期工作以来，欧几里得球的LQG体积正好适用于$ p \ in（ - \ infty，4/γ^2）$。在这里，我们证明了LQG量的LQG度量球允许所有有限的时刻。这回答了Gwynne和Miller的问题，并概括了Le Gall为Brownian Map获得的结果，即$γ= \ sqrt {8/3} $ case。我们利用这一刻来表明，在紧凑的设置上，$ r $的度量球的体积由$ r^{d_γ+o_r（1）} $给出，其中$d_γ$是LQG公制空间的尺寸。使用类似的技术，我们证明了来自公制球的Liouville Brownian运动的第一次退出时间。 Gwynne-Miller-Sheffield（2020）证明了$γ$ -LQG A.S.的度量测量空间结构当$γ= \ sqrt {8/3} $时，确定其保形结构;他们的论点和我们的估计产生了所有$γ\（0,2）$的结果。
We study the volume of metric balls in Liouville quantum gravity (LQG). For $γ\in (0,2)$, it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of Euclidean balls has finite moments exactly for $p \in (-\infty, 4/γ^2)$. Here, we prove that the LQG volume of LQG metric balls admits all finite moments. This answers a question of Gwynne and Miller and generalizes a result obtained by Le Gall for the Brownian map, namely, the $γ= \sqrt{8/3}$ case. We use this moment bound to show that on a compact set the volume of metric balls of size $r$ is given by $r^{d_γ+o_r(1)}$, where $d_γ$ is the dimension of the LQG metric space. Using similar techniques, we prove analogous results for the first exit time of Liouville Brownian motion from a metric ball. Gwynne-Miller-Sheffield (2020) prove that the metric measure space structure of $γ$-LQG a.s. determines its conformal structure when $γ=\sqrt{8/3}$; their argument and our estimate yield the result for all $γ\in (0,2)$.
