On minimal shapes and isoperimetric constants in hyperbolic lattices

We fully characterize the set of finite shapes with minimal perimeter on hyperbolic lattices given by regular tilings of the hyperbolic plane whose tiles are regular $p$-gons meeting at vertices of degree $q$, with $1/p+1/q<\frac{1}{2}$. In particular, we prove that the ratio between the perimeter and the area (i.e., the number of vertices) of this set of minimal shapes converges to the isoperimetric constant computed in H\"aggstr\"om-Jonasson-Lyons. In fact, our balls which are constructed via layers and not combinatorial balls, will realize the isoperimetric constant for any fixed number of vertices.