  In this work, we study the co-dimensional one interface limit and geometric
motions of parabolic Ginzburg--Landau systems with potentials of
high-dimensional wells. The main result generalizes the one by Lin et al.
(Comm. Pure Appl. Math., 65(6):833-888, 2012) to a dynamical case. In
particular combining modulated energy methods and weak convergence methods, we
derive the limiting harmonic heat flows in the inner and outer bulk regions
segregated by the sharp interface, and a non-standard boundary condition for
them. These results are valid provided that the initial datum of the system is
well-prepared under natural energy assumptions.
