  Monotonicity formulae play a crucial role for many geometric PDEs, especially
for their regularity theories. For minimal submanifolds in a Euclidean ball,
the classical monotonicity formula implies that if such a submanifold passes
through the centre of the ball, then its area is at least that of the
equatorial disk. Recently Brendle and Hung proved a sharp area bound for
minimal submanifolds when the prescribed point is not the centre of the ball,
which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof
involves asymptotic analysis of an ingeniously chosen vector field, and the
divergence theorem.
  In this article we prove a sharp `moving-centre' monotonicity formula for
minimal submanifolds, which implies the aforementioned area bound. We also
describe similar moving-centre monotonicity formulae for stationary
$p$-harmonic maps, mean curvature flow and the harmonic map heat flow.
