  It is well-known that the Willmore flow of closed spherical immersions exists
globally in time and converges if the initial datum has Willmore energy below
$8\pi$ - exactly the Li-Yau energy threshold below which all closed immersions
are embedded. Extending the Li-Yau inequality for closed surfaces via Simon's
monotonicity formula also for surfaces with boundary, given Dirichlet boundary
conditions, one obtains an energy threshold $C_{\mathrm{LY}}$ below which
surfaces with this boundary are embedded. By a slight modification, one obtains
a threshold $C_{\mathrm{LY}}^{\mathrm{rot}}$ below which surfaces of revolution
satisfying the boundary data have no self-intersections on the rotation axis.
  With a new argument, using this modified Li-Yau inequality and tools from
geometric measure theory, we show that the Willmore flow with Dirichlet
boundary data starting in cylindrical surfaces of revolution exists globally in
time if the energy of the initial datum is below
$C_{\mathrm{LY}}^{\mathrm{rot}}$. Moreover, given Dirichlet boundary data, we
also obtain the existence of a Willmore minimizer in the class of cylindrical
surfaces of revolution if the corresponding infimum lies below
$C_{\mathrm{LY}}^{\mathrm{rot}}$ which improves previous results for the
stationary problem.
