The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

Let gamma is an element of(0,2)ss, let h be the planar Gaussian free field, and consider the gamma\-Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a gamma-LQG metric ball with respect to the Euclidean (resp. gamma-LQG) metric is 2-gamma d gamma<mml:mfenced close=")" (>2 gamma+gamma 2+gamma 22d gamma 2 (resp. d gamma-1), where d gamma is the Hausdorff dimension of the whole plane with respect to the gamma-LQG metric. For gamma=8/3, in which case d8/3=4, we get that the essential supremum of Euclidean (resp. 8/3-LQG) dimension of a 8/3-LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and gamma-LQG Hausdorff dimensions of the intersection of a gamma-LQG ball boundary with the set of metric alpha-thick points of the field h for each alpha is an element of R. Our results show that the set of gamma/d gamma-thick points on the ball boundary has full Euclidean dimension and the set of gamma-thick points on the ball boundary has full gamma-LQG dimension.