  In this work, we study the transport of vorticity on curved dynamical
two-dimensional magnetic membranes. We find that topological transport can be
controlled by geometrically reducing symmetries, which enables processes that
are not present in flat magnetic systems. To this end, we construct a vorticity
3-current which obeys a continuity equation. This continuity equation is immune
to local fluctuations of the magnetic texture as well as spatiotemporal
fluctuations of the membrane. We show how electric current can manipulate
vortex transport in geometrically nontrivial magnetic systems. As an
illustrative example, we propose a minimal setup that realizes an
experimentally feasible energy storage device.
