  We study the electronic properties of a position-dependent effective mass
electron on a bilayer graphene catenoid bridge. We propose a position-dependent
mass (PDM) as a function of both gaussian and mean curvature. The hamiltonian
exhibits parity and time-reversal steaming from the bridge symmetry. The
effective potential contains the da Costa, centrifugal and PDM terms which are
concentrated around the catenoid bridge. For zero angular momentum states, the
PDM term provides a transition between a reflectionless to a double-well
potential. As a result, the bound states undergo a transition from a single
state around the bridge throat into two states each one located at rings around
the bridge. Above some critical value of the PDM coupling constant, the
degeneracy is restored due to double-well tunneling resonance.
