Let
be the planar
Gaussian free field and let
be a supercritical Liouville quantum gravity (LQG) metric associated with
. Such
metrics arise as subsequential scaling limits of supercritical Liouville first passage
percolation (Ding and Gwynne, 2020) and correspond to values of the matter central
charge
.
We show that a.s. the boundary of each complementary connected component of a
-metric
ball is a Jordan curve and is compact and finite-dimensional with respect to
.
This is in contrast to the
whole boundary of the
-metric
ball, which is noncompact and infinite-dimensional with respect to
(Pfeffer,
2021). Using our regularity results for boundaries of complementary connected components
of
-metric
balls, we extend the confluence of geodesics results of Gwynne and Miller (2019) to
the case of supercritical Liouville quantum gravity. These results show that two
-geodesics
with the same starting point and different target points coincide for a nontrivial
initial time interval.