Abstract
Let $$\gamma \in (0,2)$$
γ
∈
(
0
,
2
)
, let h be the planar Gaussian free field, and consider the $$\gamma $$
γ
-Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a $$\gamma $$
γ
-LQG metric ball with respect to the Euclidean (resp. $$\gamma $$
γ
-LQG) metric is $$2 - \frac{\gamma }{d_\gamma }\left( \frac{2}{\gamma } + \frac{\gamma }{2} \right) + \frac{\gamma ^2}{2d_\gamma ^2}$$
2
-
γ
d
γ
2
γ
+
γ
2
+
γ
2
2
d
γ
2
(resp. $$d_\gamma -1$$
d
γ
-
1
), where $$d_\gamma $$
d
γ
is the Hausdorff dimension of the whole plane with respect to the $$\gamma $$
γ
-LQG metric. For $$\gamma = \sqrt{8/3}$$
γ
=
8
/
3
, in which case $$d_{\sqrt{8/3}}=4$$
d
8
/
3
=
4
, we get that the essential supremum of Euclidean (resp. $$\sqrt{8/3}$$
8
/
3
-LQG) dimension of a $$\sqrt{8/3}$$
8
/
3
-LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and $$\gamma $$
γ
-LQG Hausdorff dimensions of the intersection of a $$\gamma $$
γ
-LQG ball boundary with the set of metric $$\alpha $$
α
-thick points of the field h for each $$\alpha \in \mathbb R$$
α
∈
R
. Our results show that the set of $$\gamma /d_\gamma $$
γ
/
d
γ
-thick points on the ball boundary has full Euclidean dimension and the set of $$\gamma $$
γ
-thick points on the ball boundary has full $$\gamma $$
γ
-LQG dimension.
Quasisymmetric uniformization and Hausdorff dimensions of Cantor circle Julia sets
