Asymptotic behaviour for Willmore surfaces of revolution under natural boundary conditions

where H is the mean curvature of the immersion and K its Gauss curvature. This functional models the elastic energy of thin shells. Willmore studied in [7] the functional W0, by now called Willmore functional. First we note that Wγ(Γ) ≥ 0 holds for every γ ∈ [0, 1]. Let κ1, κ2 ∈ R denote the principal curvatures of the surface. Then H − γK = 1 4 (κ1 + κ2) 2 − γκ1 · κ2 = 1−γ 4 (κ1 + κ2) 2 + γ 4 (κ1 − κ2) ≥ 0 for γ ∈ [0, 1] gives the semi-definiteness. We are interested in minima or critical points of Wγ . Such critical points have to satisfy the Willmore equation