  The multistate Potts models on two-dimensional hyperbolic lattices are
studied with respect to various boundary effects. The free energy is
numerically calculated by Corner Transfer Matrix Renormalization Group method.
We analyze phase transitions of the Potts models in the thermodynamic limit
with respect to contracted boundary layers. A false phase transition is present
even if a couple of the boundary layers are contracted. Its significance
weakens, as the number of the contracted boundary layers increases, until the
correct phase transition (deep inside the bulk) prevails over the false one.
For this purpose we derive a thermodynamic quantity, the so-called bulk excess
free energy, which depends on the contracted boundary layers and memorizes
additional boundary effects. In particular, the magnetic field is imposed on
the outermost boundary layer. While the boundary magnetic field does not affect
the second-order phase transition in the bulk if suppressing all the boundary
effects on the hyperbolic lattices, the first-order (discontinuous) phase
transition is significantly sensitive to the boundary magnetic field. Contrary
to the phase transition on the Euclidean lattices, the discontinuous phase
transition on the hyperbolic lattices can be continuously controlled (within a
certain temperature coexistence region) by varying the boundary magnetic field.
