  We investigate the formation of singularities for surfaces evolving by volume
preserving mean curvature flow. For axially symmetric flows - surfaces of
revolution - in $\mathbb{R}^3$ with Neumann boundary conditions, we prove that
the first developing singularity is of Type I. The result is obtained without
any additional curvature assumptions being imposed, while axial symmetry and
boundary conditions are justifiable given the volume constraint. Additional
results and ingredients towards the main proof include a non-cylindrical
parabolic maximum principle, and a series of estimates on geometric quantities
involving gradient, curvature terms and derivatives thereof. These hold in
arbitrary dimensions.
