  In this paper, we explore the metastable behavior of the Glauber dynamics
associated with the three-state Potts model with an asymmetrical external field
at a low-temperature regime. The model exhibits three monochromatic
configurations: a unique stable state and two metastable states with different
stability levels. We investigate metastable transitions, which are transitions
from each metastable state to the ground state, separately and verify that the
two transitions exhibit different behaviors.
  For the metastable state with greater stability, we derive large
deviation-type results for metastable transition time, both in terms of
probability and expectation. On the other hand, a particularly intriguing
phenomenon emerges when starting from the other metastable state: the process
may fall into the deep valley of another metastable state with a low
probability but remains trapped there for an exponentially long time. We
identify specific conditions on the external field under which this rare event
contributes to the mean hitting time. To this end, we conduct a detailed
analysis of the energy landscape, revealing a sharp saddle configuration
analogous to the Ising model.
