We consider the Euler-Lagrange equation of the Willmore functional coupled with Dirichlet and Neumann boundary conditions on a given curve. We prove existence of a branched solution, and for Willmore energy < 8π, we prove the existence of a smooth, embedded solution.
Mathematics Subject Classificationwhere H denotes the mean curvature vector of f, g = f * g euc the pull-back metric, µ g , µ g ∂ the induced area measures on , ∂ , respectively and κ g the geodesic curvature. The Willmore functional is scale invariant and moreover invariant under the full Möbius group of R n , see [25]. For closed surfaces, we have W( f ) ≥ 4π with equality only for round spheres, see [26] in codimension one that is n = 3, and further if W( f ) < 8π then f is an embedding by an inequality of Li and Yau [18].