论文标题
$γ$ -liouville量子重力（0,2）$中的度量胶合和共形焊接等效性
Equivalence of metric gluing and conformal welding in $γ$-Liouville quantum gravity for $γ\in (0,2)$
论文作者
论文摘要
我们考虑$γ$ -liouville量子重力（LQG）型号的$γ\ in（0,2）$，由$ e^{γH} $正式描述，其中$ h $是平面域$ d $的高斯免费字段。谢菲尔德表明，当某种类型的LQG表面（称为量子楔形物）通过适当的独立SLE曲线装饰时，将楔形物切成两个独立的表面，它们本身就是量子楔，并且可以将原始表面作为独特的保形焊接恢复。我们证明，原始表面也可以作为两个楔子的度量空间商获得，并将Gwynne和Miller的结果扩展到特殊情况下的结果$γ= \ sqrt {8/3} $ to（0,2）$。 Since the proof for $γ= \sqrt{8/3}$ used estimates for Brownian surfaces, which are equivalent to $γ$-LQG surfaces only when $γ=\sqrt{8/3}$, we instead use GFF techniques to establish estimates relating distances, areas and boundary lengths, as well as bi-Hölder continuity of the LQG metric w.r.t.边界处的欧几里得指标，这可能具有独立利益。
We consider the $γ$-Liouville quantum gravity (LQG) model for $γ\in (0,2)$, formally described by $e^{γh}$ where $h$ is a Gaussian free field on a planar domain $D$. Sheffield showed that when a certain type of LQG surface, called a quantum wedge, is decorated by an appropriate independent SLE curve, the wedge is cut into two independent surfaces which are themselves quantum wedges, and that the original surface can be recovered as a unique conformal welding. We prove that the original surface can also be obtained as a metric space quotient of the two wedges, extending results of Gwynne and Miller in the special case $γ= \sqrt{8/3}$ to the whole subcritical regime $γ\in (0,2)$. Since the proof for $γ= \sqrt{8/3}$ used estimates for Brownian surfaces, which are equivalent to $γ$-LQG surfaces only when $γ=\sqrt{8/3}$, we instead use GFF techniques to establish estimates relating distances, areas and boundary lengths, as well as bi-Hölder continuity of the LQG metric w.r.t. the Euclidean metric at the boundary, which may be of independent interest.
