  We discover new monotonicity formulae for minimal submanifolds in space
forms, which imply the sharp area bound for minimal submanifolds through a
prescribed point in a geodesic ball. These monotonicity formulae involve an
energy-like integral over sets which are, in general, not geodesic balls. In
the Euclidean case, these sets reduce to the moving-centre balls introduced by
the second author in [Zhu18].
