  In this paper we establish a gap phenomenon for immersed surfaces with
arbitrary codimension, topology and boundaries that satisfy one of a family of
systems of fourth-order anisotropic geometric partial differential equations.
Examples include Willmore surfaces, stationary solitons for the surface
diffusion flow, and biharmonic immersed surfaces in the sense of Chen. On the
boundary we enforce either umbilic or flat boundary conditions: that the
tracefree second fundamental form and its derivative or the full second
fundamental form and its derivative vanish. For the umbilic boundary condition
we prove that any surface with small L^2-norm of the tracefree second
fundamental form or full second fundamental form must be totally umbilic; that
is, a union of pieces of round spheres and flat planes. We prove that the
stricter smallness condition allows consideration for a broader range of
differential operators. For the flat boundary condition we prove the same
result with weaker hypotheses, allowing more general operators, and a stronger
conclusion: only pieces of planes are allowed. The method used relies only on
the smallness assumption and thus holds without requiring the imposition of
additional symmetries. The result holds in the class of surfaces with any genus
and irrespective of the number or shape of the boundaries.
