Rigidity result for four dimensional Willmore submanifolds with boundary

We establish a rigidity result for the critical points, with boundary, of a four dimensional Willmore energy. These critical points satisfy a 4-Willmore, equation which is a sixth order nonlinear elliptic partial differential equation. We establish several curvature estimates and, under mild assumptions on the second fundamental form, prove that four dimensional Willmore submanifolds with totally geodesic boundary condition are umbilic.