  Phase-field models such as the Allen-Cahn equation may give rise to the
formation and evolution of geometric shapes, a phenomenon that may be analyzed
rigorously in suitable scaling regimes. In its sharp-interface limit, the
vectorial Allen-Cahn equation with a potential with $N\geq 3$ distinct minima
has been conjectured to describe the evolution of branched interfaces by
multiphase mean curvature flow.
  In the present work, we give a rigorous proof for this statement in two and
three ambient dimensions and for a suitable class of potentials: As long as a
strong solution to multiphase mean curvature flow exists, solutions to the
vectorial Allen-Cahn equation with well-prepared initial data converge towards
multiphase mean curvature flow in the limit of vanishing interface width
parameter $\varepsilon\searrow 0$. We even establish the rate of convergence
$O(\varepsilon^{1/2})$.
  Our approach is based on the gradient flow structure of the Allen-Cahn
equation and its limiting motion: Building on the recent concept of "gradient
flow calibrations" for multiphase mean curvature flow, we introduce a notion of
relative entropy for the vectorial Allen-Cahn equation with multi-well
potential. This enables us to overcome the limitations of other approaches,
e.g. avoiding the need for a stability analysis of the Allen-Cahn operator or
additional convergence hypotheses for the energy at positive times.
