Minimal shapes on the hyperbolic lattices and the emergence of metastability

We investigate geometric and dynamical aspects of hyperbolic lattices arising from regular tilings with 1/p+1/q<1/2. We first characterize finite shapes with minimal perimeter and show that the ratio of perimeter to volume converges to the isoperimetric constant. We also construct a family of regular layered balls that achieve this constant for any fixed volume. We then study the Ising model on finite subgraphs with minus boundary conditions and a positive external field h. For a suitable range of h, we prove the presence of metastable behavior, identify the metastable state, and characterize the exit time. Finally, we describe the energy landscape and analyze the nucleation mechanism for all positive values of h, including beyond the metastable regime.