  Chen's flow is a fourth-order curvature flow motivated by the spectral
decomposition of immersions, a program classically pushed by B.-Y. Chen since
the 1970s. In curvature flow terms the flow sits at the critical level of
scaling together with the most popular extrinsic fourth-order curvature flow,
the Willmore and surface diffusion flows. Unlike them however the famous Chen
conjecture indicates that there should be no stationary nonminimal data, and so
in particular the flow should drive all closed submanifolds to singularities.
We investigate this idea, proving that (1) closed data becomes extinct in
finite time in all dimensions and for any codimension; (2) singularities are
characterised by concentration of curvature in $L^n$ for intrinsic dimension $n
\in \{2,4\}$ and any codimension (a Lifespan Theorem); and (3) for $n = 2$ and
in one codimension only, there exists an explicit small constant
$\varepsilon_2$ such that if the $L^2$ norm of the tracefree curvature is
initially smaller than $\varepsilon_2$, the flow remains smooth until it
shrinks to a point, and that the blowup of that point is an embedded smooth
round sphere.
